THE 

KAILKOAD   TAPER 

THE   THEORY  AND   APPLICATION  OF  A 
COMPOUND  TRANSITION  CURVE 
BASED    UPON    THIRTY- 
FOOT    CHORDS 


BY 


LEE  PERKINS 


FORMERLY  ASSISTANT  ENGINEER,  SOUTHERN  PACIFIC  LINES  IN 

ARIZONA  AND  MEXICO 
MEMBER  AMERICAN  RAILWAY  ENGINEERING  ASSOCIATION 


FIRST   EDITION 

FIRST    THOUSAND 


NEW  YORK 
JOHN  WILEY  &  SONS,  INC. 

LONDON:    CHAPMAN  &  HALL,   LIMITE] 
1915 


SV 


COPYRIGHT,  1915 

BY 
LEE  PERKINS 


Stanbope  ipress 

F.    H.GILSON   COMPANY 
BOSTON,  U.S.A. 


PREFACE 


A  VARIETY  of  transition  curves  are  used  on  the  different 
railroads,  most  of  them  fully  explained  by  published  hand- 
books. A  number  of  years  ago  Mr.  Wm.  Hood,  Chief 
Engineer  of  the  Southern  Pacific,  devised  a  compound 
transition  curve  of  unusual  simplicity,  and  developed 
several  tables  for  this  curve.  For  a  number  of  years  this 
taper  has  been  used  on  many  roads  with  somewhat  in- 
complete data,  usually  in  the  form  of  blue  prints. 

This  handbook  has  been  prepared  as  an  extension  and 
elaboration  of  Mr.  Hood's  earlier  work,  in  an  effort  to 
simplify  the  use  of  the  taper,  and  to  lessen  the  work  of  the 
man  in  the  field.  With  this  in  view,  to  a  brief  explanation 
o^  the  taper  has  been  added  the  solution  of  all  the  usual 
curve  problems  as  applied  to  tapered  curves.  Since  the 
various  transition  curves  differ  only  in  their  method  of 
development  and  their  numerical  values,  differing  not  at 
all  in  the  method  of  inserting  them  between  tangent  and 
curve,  or  between  curves,  these  curve  problems  for 
tapered  curves  are  available  for  use  with  the  tables  of  any 
transition  curve. 

The  transition  curve  herein  developed  is,  theoretically,  a 
series  of  compound  curves,  each  branch  of  the  curve  having 
a  thirty-foot  chord  (in  the  metric  system  a  ten-meter 
chord),  the  last  branch  compounding  with  the  central 
circular  curve,  the  degree  of  curvature  of  the  various 
branches  and  ending  with  the  main  curve  increasing  in 
direct  arithmetical  progression.  The  various  tables  are 
to  enable  one  to  run  in  this  series  of  compound  curves  with 

iii 

616268 


IV  PREFACE 

the  ease  of  running  in  a  simple  curve,  and  from  a  single 
setting  of  the  transit. 

The  primary  taper  tables  give  the  various  taper  func- 
tions for  every  thirty  feet  of  taper.  That  all  the  tapers 
may  be  available  for  use  with  all  degrees  and  fractional 
degrees  of  main  curve,  avoiding  the  use  of  inappropriate 
taper  lengths,  additional  tables  have  been  furnished  giving 
the  taper  functions  for  main  curves  of  intermediate  degree, 
the  values  having  been  trigonometrically  interpolated. 
If  further  interpolation  be  necessary  it  may  be  linear 
between  the  values  given  in  these  tables  of  intermediate 
values.  In  addition,  fairly  simple  equations  are  given  for 
all  the  angular  functions,  frequently  useful  in  setting 
intermediate  points  on  the  taper. 

The  radius  of  main  curve  is  assumed  based  on  a  fifty- 
foot  chord  throughout,  mainly  because  the  roads  that  have 
always  used  this  taper  have  always  used  that  radius. 
However,  this  usage  is  in  accordance  with  the  present 
tendency,  obviating  the  necessity  of  using  long  chain 
lengths  in  setting  fifty-foot  points,  which  are  set  much 
more  frequently  than  not. 

All  the  functions  of  the  taper  for  both  field  and  drawing- 
room  work  are  given  for  eleven  different  tapers  of  increas- 
ing taper  increment,  which  with  their  forty  to  eight}^ 
lengths  comprise  some  seven  hundred  tapers.  Three  of 
these  eleven,  however,  will  be  found  to  answer  most  needs 
except  in  mountainous  country. 

The  taper  numbers  used  are  those  which  have  long  been 
employed  for  these  curves,  and  are  retained  for  the  benefit  of 
present  records,  though  a  more  logical  system  of  number- 
ing, such  as  the  degree  of  curvature  increment  per  thirty 
feet  of  taper,  might  seem  desirable,  and  is  suggested  for 
use  where  this  taper  is  hereafter  adopted. 

The  problem  of  tapering  old  circular  curves  is  essentially 
a  different  one  from  locating  new  lines,  and  is  handled 
separately.  The  insertion  of  tapers  between  the  branches 


PREFACE  *  V 

of  compound  curves  should  receive  more  attention  than 
it  has  generally  had.  A  fairly  simple  solution  of  this 
problem  is  given. 

The  tapers  have  been  calculated  for  metric  curves  for 
the  use  of  the  constantly  increasing  number  of  American 
engineers  in  Mexico  and  South  America,  where  this  taper 
is  already  extensively  used. 

Acknowledgments  are  due  to  the  following,  for  the  use 
of  tables  reprinted  from  their  respective  books: 

Professor  Daniel  Carhart,  Logarithms  of  Numbers  and 
Trigonometric  and  Miscellaneous  Formulas;  Professor  J. 
C.  Nagle,  Logarithmic  Sines  and  Cosines,  Logarithmic 
Tangents  and  Cotangents,  Functions  of  a  One  Degree 
Curve,  Natural  Secants  and  Cosecants,  and  Metric  Con- 
version Table;  Mr.  Walter  Loring  Webb,  Logarithmic 
Versed  Sines  and  External  Secants,  Natural  Sines,  Cosines, 
Tangents  and  Cotangents,  and  Natural  Versed  Sines 
and  External  Secants;  and  the  American  Railway  En- 
gineering Association,  for  the  Table  of  Theoretical  Super- 
elevation of  Rails,  prepared  by  its  Committee  on  Track 
and  published  in  the  Manual  of  that  Association. 

LEE  PERKINS. 

SILVER  CITY,  NEW  MEXICO, 
March,  1915. 


CONTENTS 


CHAPTER  I 

INTRODUCTION 

PAGE 

1.  Purpose  of  the  Taper 1 

DEVELOPMENT  OF  THE  TAPER 

2.  Series  of  Compound  Curves 1 

3.  Central  Angle,  o 3 

4.  Co-ordinates  x  and  y 3 

5.  Deflection  Angles 3 

6.  Long  Chords 3 

7.  t  and  d,  giving  location  of  B.C 6 

CHAPTER  II 
LOCATION  PROBLEMS 

SIMPLE  CURVES 

8.  Semi-Tangent,  T 7 

9.  External,  E 11 

10.  Tangent  Offset  and  Distance  along  the  Tangent  to  any 

Point  on  a  Curve 11 

11.  To  Pass  a  Curve  through  a  Given  Point,  uniting  two  Given 

Tangents 11 

12.  To  Pass  a  Curve  of  Given  Radius  through  a  Given  Point, 

and  uniting  with  a  Given  Tangent 15 

13.  To  Pass  a  Curve  through  two  Given  Points  and  uniting 

with  a  Given  Tangent 16 

14.  To  fix  upon  a  Tangent  at  the  End  of  a  Curve. 18 

15.  To  Locate  a  Tangent  to  two  Located  Curves 20 

16.  To  Relocate  a  Curve  making  the  E.G.  fall  at  a  Given  Point 

in  a  Given  Parallel  Tangent f. 24 

17.  The  Intersection  of  a  Line  with  a  Curve 26 

COMPOUND  CURVES 

18.  Partial  Taper  between  the  Branches  of  the  Compound 

Curve 27 

vii 


Vlll  CONTENTS 

PAGE 

19.  Taper  Functions  of  the  Partial  Taper 28 

20.  To  find  the  Semi-Tangents,  both  Radii  and  both  Central 

Angles  given 31 

21.  To  Compound  a  Simple  Curve  so  the  E.G.  shall  fall  in  a 

Given  Parallel  Tangent,  both  Radii  given 33 

22.  To  Compound  a  Simple  Curve  so  the  E.G.  shall  fall  in  a 

new  Located  Tangent,  both  Radii,  or  both  Central  Angles 
given 34 

23.  To  find  the  Second  Semi-Tangent  and  the  Central  Angles, 

one  Semi-Tangent,  the  Intersection  Angle  and  both  Radii 
given 36 

CURVES  BEYOND  THE  FROG 

24.  Taper  beginning  at  the  Heel  of  Frog 37 

CHAPTER  III 
FIELD  WORK 

25.  Selection  of  the  Taper 40 

26.  Laying  out  a  Tapered  Curve 41 

OBSTRUCTIONS   ON  THE  TAPER 

27.  Curve  run  from  Main  Curve  Produced 42 

28.  Curve  run  from  a  Point  beyond  the  C.C 44 

29.  Taper  Points  set  by  Offset. ..', 45 

30.  Taper  Points  set  from  Point  on  Semi-Tangent 45 

31.  Curve  Notes  for  the  Field  Book 46 

CHAPTER  IV 
OFFICE  WORK 

32.  Plotting  a  Tapered  Curve 47 

33.  Paper  Location  with  Tapers 47 

34.  Curve  Compensation  with  Tapers 47 

35.  Right  of  Way  Descriptions  with  Tapers 48 

CHAPTER  V 
TAPERING   OLD  TRACK 

36.  Condition  of  Track 49 

37.  Operating  Conditions 50 


CONTENTS  IX 

METHODS   OF  INSERTING  TAPERS 

SIMPLE  CURVES 

PAGE 

38.  Shifting  the  Curve 50 

39.  Shifting  the  Tangents 52 

40.  Shifting  Curve  and  Tangents 53 

41.  Increasing  Degree  of  Curve 53 

42.  Increasing  Degree  and  Shifting  Curve 54 

43.  Maintaining  Length  of  Line  Unchanged 55 

44.  Increasing  Degree  and  Shifting  Curve  and  Tangents 57 

45.  Combination  of  Methods 57 

46.  Increasing  Degree  of  Portion  of  Curve 59 

COMPOUND  CURVES 

47.  Similarity  of  Procedure 59 

48.  Shifting  the  Curves 59 

49.  Shifting  Curves  and  Tangents 60 

50.  Shifting  Lighter  Branch  and  Increasing  Degree  of  Sharper 

Branch 60 

51.  Field  Work  with  Compound  Curves 61 

TABLES 

Radii  of  Curves  and  their  Logarithms 63 

Notation  used  in  the  Taper  Tables 69 

Important  Equations  abstracted  from  the  Text 69 

Taper    £,  A  =    1 71 

1  ,A  =   'J 74 

U,  A  =     f 78 

2  ,  A  =  1 82 

21,  A  =  11 86 

2i  A  =  U - 90 

2f,  A  =  2 94 

3  ,  A  =  2£ 97 

3i,  A  =  3 100 

3£,  A  =  4 103 

3,f  A  =5 105 

Comparative  Values  of  I,  d,  and  o V 108 

Maintaining  Length  of  Line  Unchanged 115 

Logarithms  of  Numbers 117 

Logarithmic  Sines  and  Cosines 135 

Logarithmic  Tangents  and  Cotangents 150 

Logarithmic  Versed  Sines  and  External  Secants 165 


X  CONTENTS 

PAGE 

Functions  of  a  One  Degree  Curve 210 

Natural  Sines,  Cosines,  Tangents  and  Cotangents 240 

Natural  Secants  and  Cosecants 263 

Natural  Versed  Sines  and  External  Secants 276 

METRIC   TABLES 

Radii  of  Curves  and  their  Logarithms 299 

Notation  used  in  the  Taper  Tables 305 

Important  Equations  abstracted  from  the  Text 305 

Taper    £,  A  =    £ 307 

1  ,  A  =    i 310 

U,  A  =    f 313 

2  ,  A  =  1    317 

2i  A  =  H. 321 

2£,  A  =  H 325 

2f,  A  =  2 329 

3  ,  A  =  2^ 333 

3J,  A  =  3 337 

3£,  A  =  4 340 

3f ,  A  =  5 342 

Comparative  Values  of  I,  d,  and  o 344 

Mutual  Conversion  of  Feet  and  Inches  into  Meters  and  Centi- 
meters   351 

Trigonometric  and  Miscellaneous  Formulas 352 

Super-elevation  of  Rails  on  Curves , 355 


THE   RAILROAD  TAPER 


CHAPTER  I 
INTRODUCTION 

PURPOSE  OF  THE  TAPER 

1.  Tapered  curves  have  gained  such  wide  us^age  that 
it  is  needless  to  dwell  at  length  on  their  advantages. 
Where  circular  curves  alone  are  used,  the  transition  from 
tangent  to  curve  is  sudden,  giving  a  decided  lurch  to  the 
cars,   unpleasant  to  passengers  and   damaging  alike  to 
rolling   stock   and   track.     Running   the   super-elevation 
back  on  the  tangent  does  not  obviate  this,  and  it  can  be 
remedied  only  by  increasing  the  curvature  gradually,  in- 
creasing the  super-elevation  of  rails  at  the  same  time. 

This  the  tapered  curve  accomplishes  in  a  satisfactory 
and  practical  manner,  much  as  it  has  always  been  [done  by 
track  foremen,  who  lined  in  an  approximate  taper  by  eye. 
The  proper  use  of  the  taper,  however,  insures  a ,  uniform 
curve  beyond  the  taper,  a  result  not  easily  obtained  by  the 
trackman  unaided. 

DEVELOPMENT  OF  THE  TAPER 

2.  The  railroad  taper  consists,  theoretically,  of  a  series 
of  short  curves,  each  of  thirty-foot  chord  lengttj,  and  of 
uniformly  increasing  curvature,  compounding  at:;  the  end 
with  the  central  main  curve.     The  various  tapirs  differ 
merely  in  the  curvature  increment,  and,  therefore,  in  the 
length  needed  to  lead  up  to  the  curvature  of  the  main 
curve.     Fig.  1  illustrates  the  first  one  hundred  and  twenty 

1 


THK  RAILROAD  TAPER 


DEVELOPMENT  OF  THE  TAPER  3 

feet  of  Taper  2,  compounding  with  the  main  five-degree 
curve. 

3.  The  central  angle  (o)  of  the  taper  is,  of  course,  the 
summation  of  the  central  angles  of  its  parts. 

4.  In  Fig.  2  is  shown  the  same  curve  represented  by 
its  thirty-foot  chords,  showing  the  method  of  obtaining 
the  x  and  y  co-ordinates  of  the  chord-points  referred  to 
the  initial  tangent  and  the  B.C.  (beginning  of  curve)  of  the 
taper.     The  angle  k  is  the  azimuth  of  the  various  chords, 
referred  to  the  initial  tangent,  and  the  x  and  y  co-ordinates 
are  the  departures  and  latitudes  of   the   chord-points. 
These  co-ordinates  are  given  in  the  tables. 

5.  From  x  and  y  were  calculated  for  the  tables  the  deflec- 
tion angles  (/)  from  the  initial  tangent  at  the  B.C.  to  the 
various  chord-points,  by  the  equation 

tan/  =  I-  (1) 

y 

The  backsight  deflection  (6)  from  tangent  at  any  chord- 
point  to  the  B.C.  equals  the  central  angle  of  the  taper 
up  to  that  point  minus  the  deflection  at  the  B.C.  to  that 
chord-point,  and  is  given  in  the  tables. 

As  shown  in  Fig.  3,  the  deflection  from  tangent  at  any 
chord-point  to  any  other  chord-point  equals  the  azimuth 
of  the  long  chord  between  the  chord-points,  minus  the 
central  angle  of  the  taper  up  to  the  first  chord-point. 

x"  —  x' 
This  azimuth  is  the  angle  whose  tangent  is  -77—  —  -,- 

y       y 

As  shown  in  Fig.  4,  the  backsight  deflection  from  tangent 
at  any  chord-point  to  any  other  chord-point  is 


and  is  given  in  the  tables. 

6.   The  long  chord  (Ic)  from  the  B.C.  to  any  chord- 
point,  given  in  the  tables,  is 

Ic  =  -^-,-  (2) 

cos/ 


THE  RAILROAD  TAPER 


FlQ.  3. 


FIG.  4. 


FIG.  6. 


(5) 


0  THE   RAILROAD  TAPER 

Similarly,  as  in  Fig.  5,  the  long  chord  between  any  other 
chord-points  is 

IP'-"  -     y  —  y  o\ 


cos  (e'  +/'-") 

7.  In  order  to  make  room  for  the  taper,  the  simple  curve 
may  be  considered  as  offset  from  its  natural  position  a 
distance  from  the  tangent  that  will  be  called  d,  the  taper 
beginning  at  a  point  back  of  this  offset  position  of  the 
B.C.  of  the  simple  curve  a  distance  along  the  tangent  that 
will  be  called  t. 

This  tangent  distance,  t,  is  very  nearly  half  the  length 
of  the  taper,  its  exact  value,  from  Fig.  6,  being 
t  =  BD  -  IE 
=  y  —  R  sin  o,  (4) 

where  R  is  the  radius  of  the  main  curve,  and  o  the  central 
angle  of  the  taper. 

The  value  of  the  circular  curve  offset,  d,  from  Fig.  6,  is 
d  =  DE  -  HI 
=  x  —  R  vers  o.  (5) 

The  values  of  t  and  d  are  given  in  the  tables. 

These  two  equations,  4  and  5,  are  the  fundamental 
equations  of  the  taper,  locating  the  center  of  curvature 
of  the  main  curve  with  reference  to  the  tangents  and  the 
beginning  and  end  of  curve.  The  equations  are  identical 
in  form  for  all  transition  curves,  and  similar  values  under 
different  letter  designations  are  invariably  given  in  their 
tables.  On  these  two  distances,  t  and  dy  are  based  all  the 
solutions  of  curve  problems  in  the  succeeding  chapters. 


CHAPTER  II 
LOCATION   PROBLEMS 

8.  To  Find  the  Semi-tangents  of  a  Tapered  Curve. 
Given  R  and  /  to  find  T. 


FIG.  7. 

CASE  I.     Tapered  at  both  ends  with  the  same  taper. 
From  Fig.  7, 

T  =  AD  +  AB 
=  t  +  OA  tan  J  7 

=  «  +  (#  +  d)  tan  1 7.  (6) 

7 


8 


THE   RAILROAD  TAPER 


In  the  tables  is  given  a  value  D  =  R  +  d,  so  the  above 
may  be  written 

T  =  t  +  D  tan  }  7.  (7) 

Example.     Find  the  semi-tangent   of  a  5°  00'   curve, 
tapered  with  Tap.  2,  total  angle  of  curve  35°  04'. 

By  Eqn.  7      log  D5°  Tap  2          3 . 05949 

log  tan  17°  32'      9.49960 

2.55909 

362.32 

t  =      59.99 

T  =    422.31 

CASE  II.     Tapered  at  one  end  only. 
It  may  occasionally  be  necessary  to  taper  one  end  only 
of  a  curve,  or,  considering  the  parallel  offset  tangent,  to 
treat  the  curve  as  tapered  at  one  end  only. 


FIG.  8. 

In  this  case  the  two  semi-tangents  will  be  of  different 
lengths,  the  one  for  the  tapered  end  being  the  longer,  when 
I  is  less  than  90°,  by  an  amount  less  than  t. 

From  Fig.  8. 

MN  =  MK  =  dcscI, 
Tt  =  t  +  OAtau±I  -  MN 

=  t  +  D  tan  J  /  -  d  esc  I,  (8) 

Tn  =  R  tan  J I  +  d  esc  7.  (9) 


LOCATION  PROBLEMS  9 

These  equations  do  not  change  for  angles  greater  than 
90°,  the  algebraic  sign  of  the  last  term  remaining  un- 
changed. 

Example.  Find  the  semi-tangents  of  a  7°  00'  curve, 
tapered  at  the  B.C.  with  Tap.  2J,  no  taper  at  the  E.C.t 
total  angle  of  curve  54°  22'. 

By  Eqn.  8      esc  54°  22'  1 . 23037 

d  0.95 

615185 
1  007333 


1.0688515 

2|  2.91360 

Iogtan27°ll/      9.71059 
2.62419 

420.91 

+  t     54.98 

475.89 

1.07 


B.C.  to  P.  I.  474.82 

By  Eqn.  9      log  Rr  2.91309 

log  tan  27°  11'      9.71059 
2.62368 

420.42 
+  1.07 


P.I.  to  E.G.     421.49 

CASE  III.     Tapered  at  both  ends,  with  different  tapers. 
From  Fig.  9, 

MN  =  MK  =  dlcseI, 
BM  =  dz  esc  I, 
BJ  =  BN  =  MN  -  BM 

=  (di  —  d%)  esc  /, 

Ti  =  ^  +  A  tan  J  /  -  (di  -  4)'  esc  7.         (10) 

T2  =  t*  +  I>2  tan  J7  -  (dz  -  di)  esc/.         (11) 

These  equations  do  not  change  for  angles  greater  than 

90°,  the  algebraic  sign  of  the  last  term  remaining  un- 

changed. 


10 


THE  RAILROAD  TAPER 


Example.  Find  the  semi-tangents  of  a  10°  00'  curve, 
tapered  at  the  B.C.  with  Tap.  2i,  and  at  the  E.G.  with 
Tap.  3,  total  angle  of  curve  113°  42'. 


-tr- • 


By  Eqn.  10 


FIG.  9. 

esc  113°  42' (esc  66°  18')  1.09211 
4  -  d,  =  2.84  -  0.98  =       1.86 

655266 
873688 
1  09211 


log  DIO°  Tap.  2i 

log  tan  56°  51 


By  Eqn.  11 


B.C.  to  P.I. 

log  DIO°  Tap.  3 

log  tan  56°  51 


2.03 

2.76041 
0.18500 
2.94541 

881.88 

84.88 

966.76 

2.03 

964.73 

2.75900 
0.18500 
2.94400 


LOCATION  PROBLEMS  11 

879.02 

+fc  44.97 

923.99 

+  2.03 

P. I.  to  E.G.  926.02 

9.  Given  M  and  J,  to  find  the  External,  E. 
CASE  I.     Tapered  at  both  ends  with  the  same  taper. 
From  Fig.  7, 

E  =  OB  -  R 

=  D  sec  j  7  -  #,  (12) 

or  '    E  =  (72  +  d)  sec  J  7  -  #  -  d  +  d 

=  (Jf2  +  d)  (sec  J  7  -  1)  +  d 
=  D  exsec  J  7  +  d.  (13) 

Example.     Find  the  external  of  a  5°  00'  curve,  Tap.  2, 
total  angle  27°  12'. 

By  Eqn.  12  or  13,        E  =  33.88. 

CASE  II.     Tapered  at  one  end  only,  or  at  both  ends,  with 
different  tapers. 
From  Fig.  10, 

E  =  VOA2  +  AB2-R 

=  VD*  +  (T  -  o2  -  R,  (14) 

or  E  =  D  sec  tan-1  ^^  -  R,  (15) 

T  —  t  T  —  t 

where  tan"1  — ^ —  is  the  angle  whose  tangent  is  — j? — 

Example.     Find  the  external  to  the  curve  given  in  the 
example  under  equations  8  and  9. 
By  Eqn.  14  or  15,       E  =  102.24. 

10.  To  Find  the  Tangent  Offset  and  Tangent  Distance 
to  any  Point  on  the  Curve. 

From  Fig.  11, 

PD  =  AC  +  d 

=  RversI  +  d,  (16) 

ED  =  BD  +  t 

=  R  sin  /  +  t.  (17) 


12 


THE  RAILROAD  TAPER 


FIG.  10. 


FIG.  11. 


LOCATION  PROBLEMS 


13 


Or  from  the  Table  of  Functions  of  a  1°  curve,  AC  is  the 
mid-ordinate  for  a  central  angle  equal  to  2  /,  and  BD  is 
half  the  long  chord  for  a  similar  central  angle. 

11.  Given  a  Point,  P,  referred  to  the  JP.J.,  to  find 
the  Radius  of  a  Curve  passing  through  F  uniting  the 
Given  Tangents. 

Since  the  solution  of  this  problem  depends  upon  the 
taper  chosen  and  its  value  of  d,  and  the  choice  of  taper 
depends  largely  upon  the  degree  of  curve,  a  trial  solution 
for  an  untapered  curve  is  first  made. 


From  Fig.  12, 


FIG.  12. 


OB   . 

sin  y  =         sin  a 


R  sec  \  I   . 

-TT-sina 

sin  a 


cosf 
R  =  PO 
sin  a 


sinx 


PB. 


(18) 
(19) 


14 


THE   RAILROAD  TAPER 


Since  this  problem  occurs  mainly  in  location,  and  it  is 
necessary  merely  to  have  the  curve  pass  close  to  the 
determining  point  without  using  a  curve  of  fractional 
degree,  to  allow  room  for  the  taper  a  curve  of  next  higher 
even  degree  is  chosen,  and  when  desirable  it  may  be 
determined  as  follows  how  close  the  chosen  curve  will  pass 
to  the  given  point. 

In  Fig.  13,  P  is  the  point  through  which  the  curve  should 
pass,  Pr  a  point  on  the  chosen  curve  radially  opposite. 

PP'=QP-R 


=     (OA  -  AD)2  +  (AB  -  CB)2  -  R 

=  V(D-PB  sin  6)2+  (D  tan  J  /  -  PB  cos  6)2  -  R.  (20) 


FIG.  13. 

Example.     In  Fig.  13,    let  PB  =  194.3,    b  =  42°  24', 

^4°  22'  r 

I  =  54°  22'.     Then    o  =  90°  -      2       -  42°  24' 

=  20°  25'. 

By  Eqn.  18,  y  =  23°  05' 

x  =  y-a=    2°  40', 

By  Eqn.  19,  R  =  1456.8. 

3°  56'  curve. 


LOCATION  PROBLEMS 


15 


Deciding  now  upon  a  4°  00'  curve  with  Tap.   1,   to 
determine  how  close  this  curve  will  pass  to  point  P: 
By  Eqn.  20,     (D  -  PB  sin  6)2  =?  1698100 
(D  tan  %  I-PB  cos  6)2=  351675 
2049775 

V2049775  =   1431.7 
-  R  1432.5 

PP'       =   ~  -  0.8 

The  negative  result  shows  that  P  is  inside  the  curve 
0.8  ft.,  which  on  most  location  is  sufficiently  close. 

12.  Find  the  B.C.  of  a  Curve  of  Given  Radius  that  will 
pass  through  a  Given  Point  referred  to  a  Given  Tangent. 


\ 


FIG.  14. 


From  Fig.  14, 


AB  =  R  sin  vers"1 


BP-d 
D 


(21) 


f  R  times  the  sine  of  the  angle  whose  versine  is  -    ~      J 
or  R2=  (D-  BP)* 


AB  = 


-  (D-  BP)2. 


(22) 


16 


THE  RAILROAD  TAPER 


Example.    Given  BP  =  82  ft.,   M  =  6°  00',   Tap.   2, 
then    D  =  956.41,     R  =  955.04. 

r>p   7 

By  Eqn.  21,    vers-1  -~  -  =  22°  45J', 


3G9.45. 


13.  Given  Two  Points  referred  to  a  Tangent,  to  find 
the  Radius  and  B.C.  of  a  Curve  passing  through  them 
and  joining  the  Given  Tangent  with  a  Given  Taper. 

Since  the  solution  of  this  problem  depends  upon  the 
taper  chosen  and  its  value  of  d,  the  choice  of  which  depends 
largely  upon  the  degree  of  curve,  a  trial  solution  for  an 
untapered  curve  is  first  made. 


FIG.  15. 

In  Fig.  15,  given  points  P  and  S,  referred  to  the  tangent 
by  lengths  DP  and  DS,  and  the  angle  a,  to  find  the  radius 
OA  and  distance  AD. 


By  geometry       AD2  =  DP  X  DS, 


AD  =  VDP  X  DS. 


(23) 


DB  and  BP  may  be  obtained  from  a  solution  of  the  tri- 
angle BDP. 


LOCATION  PROBLEMS 


17 


D 

£1   = 


=  AB2  +  (R  -  BP)2 
=  AB2  +  R2 
AB2  +  BP2 


BP2, 


or 


2BP 
AB 


sin  2  tan"1  -r 


(24) 
(25) 


A  curve  of  next  higher  even  degree  is  now  chosen  with 
an  appropriate  taper,  passed  through  S  as  in  paragraph 
12,  and  the  distance  of  P  from  the  curve  determined  as 
follows. 


FIG.  16. 
From  Fig.  16, 

A'C  =  R  sin  vers"1 


or 


-  (D  -  CS)2', 
PP'  =  OP-R 

=  VOE2  +  EP2-R      ' 


or 


PP'  = 


=  V(D  -  BP)2  +  A'B2-  R, 
D-BP 


cos  tan~ 


A'B 
D-BP 


-R. 


(26) 
(27) 

(28) 
(29) 


18  THE  RAILROAD  TAPER 

Example.    DP  =  94  ft.,  PS  =  127  ft.,  angle  a  =  16°  12'. 
By  Eqn.  23,  AD  =  144.1. 

FromABDP,  BP  =    26.2 

and  DB  =    90.3; 

then  AB  =  AD  +  DB  =  234  .  4. 

7?P 
By  Eqn.  25,       tan-1  =%  =  6°  23', 


and  #  =  1060.7 

5°  24'  curve. 

Deciding  now  upon  a  5°  30'  curve,  Tap.  2,  through  S: 
From  A  DSC,  CS  =    61.66 

-  d          1.05 
CS-d=    60.61 

and  DC  =  212.2 

-DB        90.3 
#C  =  121.9. 

By  Eqn.  26,  vei&-lCS~d  =  19°  38' 
tt 

and  A'C  =  350.06 

-BC       121.9 
A'B  =  228.16. 

By  Eqn.  29,    tan'1  =  12°  39', 


and  PP'  =  0.14. 

From  the  positive  value  of  PP',  P  lies  outside  the  curve 
assumed.  If  the  curve  had  not  passed  sufficiently  close 
to  P,  a  slightly  sharper  curve  would  pass  further  out  at 
P',  or  if  the  curve  had  passed  outside  of  P,  a  slightly 
lighter  curve  would  pass  closer  to  P. 

14.  To  Fix  upon  a  Tangent  at  the  End  of  a  Curve. 

In  the  previous  problem  of  passing  the  curve  through 
two  determining  points  and  tangent  to  the  back-located 
tangent,  it  is  not  practical  to  first  locate  the  tangent 


LOCATION  PROBLEMS 


19 


ahead  and  make  an  intersection.  The  curve  must  be 
run  in  beyond  the  determining  points,  then  a  meander  line 
run  to  the  next  controlling  point  on  the  tangent  ahead 
and  the  tangent  located  as  follows: 

When  the  curve  has  reached  a  point  A,  Fig.  17,  from 
which  a  view  may  be  had  of  the  tangent  desired,  or  of 
some  controlling  point,  B,  on  it,  through  which  it  is 
desired  to  run  the  tangent  because  of  properly  supporting 
ground,  a  flag  may  be  sent  to  C,  opposite  B,  (the  uncer- 


FIG.  17. 

tainty  of  right  angle  direction  will  affect  this  very  little), 
and  a  distance  therefrom  equal  to  d  of  the  taper.  With 
improvised  targets  of  some  nature,  or  double  points  and 
angle  between,  a  fairly  accurate  stadia  shot  may  be 
obtained,  and  with  the  angle  from  tangent  at  A  to  C,  the 
distance  BL  and  the  central  angle  of  the  rest  of  the  curve 
may  readily  be  calculated. 
From  Fig.  17, 

AG  =  2R  sin  a, 

OK  =  R  cos  a, 

CF  =  VCG  X  AC, 
OK 


(30) 


Rcosa 
AC  —  Rsina 


20  THE   RAILROAD  TAPER 


R 

tan  m  = 


R  (3D 


V(AC  -2Rsma)AC 
b  =  a  +  m  —  n.  (32) 

Given  b,  AD  and  Z)L  may  be  calculated  by  equations 
9  and  8.  Then 

LB  =  CF  -t  

=  V(AC  -  2  #  sin a)  AC  -  t.     (33) 

Example.  Assume  a  5°  30'  curve,  Tap.  2,  to  have  been 
run  to  point  A.  Distance  AC  by  stadia  is  610  ft.,  and 
angle  a  is  6°  10'. 

By  Eqn.  30,  n  =  64°  19'. 

By  Eqn.  31,  m  =  66°  01'. 

By  Eqn.  32,  a        6°  10' 

+  ra      66°  01' 
-  n      64°  19' 
6=7°  52'. 
By  Eqn.  33,  LB  =  417.9. 

15.   To  Connect  two  Located  Curves  by  a  Tangent. 

In  the  preceding  problem,  if  the  next  controlling  point 
had  been  on  a  curve,  instead  of  a  tangent,  it  would  often 
be  necessary  to  locate  this  next  curve  first  and  then  con- 
nect the  two  curves  by  a  tangent. 

CASE  I.     Curves  of  contrary  flexure. 

Let  AB  and  CD,  Fig.  18,  be  the  two  located  curves,  to 
locate  tangent  LM  fitting  the  two  curves  with  given 
tapers,  the  curves  being  offset  from  the  tangent  di  and  d2, 
respectively,  to  provide  room  for  the  tapers. 

From  some  point  E  on  tangent  at  B,  run  one  or  more 
courses,  EF,  to  some  point  F  on  tangent  at  D.  Calculate 
the  latitude  and  departures  of  N  and  0  with  reference  to 
tangent  at  B  as  base  and  radius  BN  as  meridian.  The 
difference  of  the  latitudes  equals  OG,  the  difference  of  the 
departures  equals  NG. 


LOCATION  PROBLEMS 


21 


Then, 


NO  =  VOG2  +  NG2, 
OG 


+  A  +  #2  + 


cos  c 


a  =  90°  -  e  -  c, 
/  _  g,  =  a  _  6, 

tt-f«n/\*A 

LM  =  ST  -  (t,  + 
=  NO  sine- 


(34) 
(35) 

(36) 
(37) 

(38) 
(39) 


22 


THE  RAILROAD  TAPER 


Example.  Given  a  5°  30'  curve  AB  and  a  4°  00'  curve 
CD,  to  locate  tangent  connecting  them,  using  Taper  2 
on  the  5°  30'  curve  and  Taper  l£  on  the  4°  00'  curve. 

At  B  turn  tangent  to  curve  and  set  point  E,  and  at  D 
turn  tangent  and  set  F.  Assume  BE  =  100  ft.,  angle 
/  =  9°  11',  EF  =  384.0  ft.,  angle  g  =  6°  38',  FD  =  100  ft. 

Then  a  traverse  from  N  to  0  is  as  follows: 


Course. 

Length. 

Azimuth. 

Lat. 

Dep. 

NB  . 

1041.8 

180°  00' 

-1041.8 

0 

BE.. 
EF.. 

100.0 
384.0 

90°  00' 
80°  49' 

0 
+     61.3 

+  100.0 
4-379.1  - 

FD  
DO 

100.0 
1432  5 

87°  27' 
177°  27' 

+      4.4 
-1431.1 

+  99.9 
+  63.7 

NO  

2491.5 

165°  03' 

-2407.2 

+642.7 

From  this  traverse 


NO  =  2491.5, 

.  NG  =    642.7, 

GO  =  2407.2, 

e  =  75°  03'. 

By  Eqn.  36, 

c  =  6°  23'. 

By  Eqn.  37, 

a  =  8°  34'. 

By  Eqn.  38, 

6  =  6°  01'. 

By  Eqn.  39, 

LM  =  144.5. 

By  Eqn.  9, 

BH  =    85.08. 

By  Eqn.  8, 

HL  =  138.54. 

By  Eqn.  8, 

MK  =  133.54. 

By  Eqn.  9, 

KD  =    82.05. 

If  in  this  solution  LM  had  been  too  short,  or  even 
negative,  one  of  the  curves  would  have  to  be  shifted  or 
sharpened,  or  the  tapers  shortened. 

CASE  II.     Curves  of  similar  flexure. 
Determine  NG  and  GO  as  in  Case  I. 


LOCATION  PROBLEMS 


A  FIG.  19. 

Then  from  Fig.  19, 


NO  =  VOG2  +  NG2, 

OG 
i^e  =  m, 


cos  c  = 


a  =  e  +  c  -  90°, 
6=/  +  <7-a, 
LM  =  NOsmc-  (ti 


24 


THE   RAILROAD  TAPER 


Example.  Assume  the  same  conditions  as  in  the 
example  under  Case  I,  angle  g  merely  being  in  the  opposite 
direction. 

Then  a  traverse  from  N  to  0  is  as  follows: 


Course. 

Length. 

Azimuth. 

Lat. 

Dep. 

NB 

1041  8 

180°  00' 

-1041.8 

0 

BE 

100  0 

90°  00' 

0 

+  100  0 

EF.. 

384.0 

80°  49' 

+     61.3 

+379.1 

FD 

100  0 

74°  ir 

+     27  3 

+  96  2 

DO 

1432  5 

344°  IT 

+  1378  3 

-390  4 

NO  

485.4 

23°  30' 

+  425.1 

+184.9 

From  this  traverse 


By  Eqn.  36a, 
By  Eqn.  37a, 
By  Eqn.  38a, 
By  Eqn.  39, 


NO  =  485. 4, 

e  =  90°  -  23°  30' 
=  66°  30'. 

c  =  36°29'. 

a  =  12°  59'. 

6=2°  50'. 
LM  =  156.1. 


16.  To  Relocate  a  Curve  so  the  E.C.  shall  fall  at  a 
Given  Point  in  a  Given  Parallel  Tangent. 

In  Fig.  20  let  ABC  be  the  located  curve,  tapers  omitted 
for  clearness,  and  DEF  the  required  curve  ending  at  F. 
CG  and  GF  are  known. 

By  a  solution  of  triangle  KM  I,  KF  becomes  known. 
The  value  of  R2  now  depends  solely  upon  the  taper  chosen 
and  its  value  of  t%  and  d2,  and  may  best  be  determined  by 
trial,  using  the  tables  of  functions  of  a  one-degree  curve, 
assuming  at  first  a  value  of  fe  somewhat  larger,  or  smaller 
as  the  case  may  be,  than  t\.  It  will  usually  be  sufficient 
for  the  semi-tangent  of  the  new  curve  approximately  to 
equal  KF. 


LOCATION  PROBLEMS 


25 


Example.     Assume  curve  ABC  to  be  a  6°  00'  curve, 
Taper  2,  total  angle  36°  14'.     CG  =  7.1,     GF  =  42.4. 

K      I 


FIG.  20. 

From  A  KM  I,          KM  =      9.7 
and  KI  =    12.0. 

By  Eqn.  7,  1C  =  387.88 

+     KM        9.7 
+      GF       42.4 
KF  =  440.0. 
From  Tables,  1°  curve 

TWW  =  1874.7. 

If  the  new  curve  were  untapered, 
1874.7 


M 


4.26°. 


440.0 
If  Taper  2  be  used  on  the  new  curve,  then 


M 

1847.7 
M 

\l 

Apprbx.  T. 

4°  45' 

394.7 

56.2 

450.9 

50' 

387.9 

57.5 

445.4 

55' 

381.3 

58.7 

440.0 

5°  00' 

374.9 

60.0 

434.9 

26 


THE  RAILROAD  TAPER 


Assuming  a  4°  55'  curve,  Taper  2,  then 
by  Eqn.  7,  T2  =  440.29. 

In  this  particular  case  it  is  unnecessary  to  interpolate 
to  obtain  a  more  exact  value  for  the  semi-tangents,  0.29  ft. 
being  sufficiently  close.  Linear  interpolation  between  the 
intermediate  values  given  in  the  tables  would  be  quite 
accurate  when  necessary. 

17.  Intersection  of  a  Line  with  a  Curve. 
In  Fig.  21,  given  R,  d,  AB  and  angle  6,  to  find  BP  and 
angle  a. 


tan  n  = 


FIG.  21. 


D 


AB' 

c  =  180°  -  n  -6, 
'  =  n  +  6'  -  180°, 


sin  n 

OB  . 
sin  e  =  —^  sin  c 

£1 


LOCATION  PROBLEMS  27 

(42) 


n 

=  (e-c)  (43) 

sine 
«=/+</ 

=  90°  -  TO  +  e  -  c,  (44) 

<*'=/-</' 

=  90°  -  n  +  c'  -  e'.  (44a) 

In  equations  42  and  43,  c'  may  be  substituted  for  c  with 
no  other  changes. 

Example.  Given  a  4°  00'  curve,  Taper  1  J,  A£  =  195.9 
ft.,  6  =  67°  07'. 

By  Eqn.  40,  n  =  82°  13'. 

By  Eqh.  41,  c  =  30°  40'. 

By  Eqn.  42,  e  =  31°  00'. 

By  Eqn.  43,  BP  =  16.34. 

By  Eqn.  44,  a  =    8°  07'. 

COMPOUND  CURVES 

18.  A  Taper  between  the  Branches  of  a  Compound 

Curve  is  often  more  necessary  than  between  curves  and 
tangents,  as  the  change  in  curvature  is  often  greater. 
Only  that  portion  of  the  complete  taper  required  to  gradu- 
ally increase  the  curvature  as  desired,  is  used. 

Example.  Taper  2  from  tangent  to  a  6°  00'  curve  is 
built  up  from  30  ft.  of  1°,  30  ft.  of  2°,  30  ft.  of  3°,  30  ft.  of 
4°  and  30  ft.  of  5°  curves,  but  in  compounding  from  a  2° 
to  a  6°  curve  with  Taper  2,  the  first  60  ft.  is  dropped  off, 
the  deflection  tables  being  used  as  if  the  transit  were  at 
/  =  60,  the  length  of  taper  to  fit  a  curve  of  M°  =  MI  +  A, 
where  MI  is  the  degree  of  the  lighter  branch,  and  A  the 
curvature  increment  in  degrees  for  the  taper  used. 


28  THE   RAILROAD  TAPER 

19.   Taper  Functions  of  the  Partial  Taper. 

From  the  method  of  developing  the  taper,  the  length  of 
any  taper  from  tangent  is, 

J  =  J(M-A).  *(45) 

Then  the  entire  taper  from  tangent  to  fit  the  branch  of 
the  compound  curve  of  greater  curvature  is 

OA 

/2  =  ^(M2-A),  *(46) 

and  the  length  replaced  by  the  branch  of  lesser  curvature 
is  that  needed  to  fit  a  curve  of  M°  =  M  i  +  A  degree,  or 


Then  the  partial  taper  retained  between  the  branches 
of  the  compound  has  a  length  of 


=       (M2-M!-A).  *(48) 

In  Fig.  22,  Mi  is  the  curve  of  lesser  degree,  M2  the  curve 
of  greater  degree,  and  ABC  the  taper  joining  the  two 
curves,  the  tapers  at  the  beginning  and  end  of  the  com- 
pound curve  being  omitted  for  clearness  of  diagram. 

The  two  curves  are  offset  from  each  other  to  make  room 
for  the  taper  an  amount  KF  =  dr,  radially  to  both  curves, 


*  Equations  45,  46,  47,  48,  50,  51a,  56  and  57  are  empirical 
equations  derived  from  the  tables,  and  cannot  be  used  with  other 
transition  curves.  When  the  taper  functions  of  Mi  and  M2  are 
given  in  the  tables  these  equations  are  not  needed.  When  needed 
for  other  transition  curves,  somewhat  similar  equations,  but  with 
other  constants,  must  be  derived. 


LOCATION  PROBLEMS 


29 


the  curve  of  greater  curvature  lying  within  the  one  of 
lesser  curvature. 

dr  =  d  for  M  =  M2  -  Ml  (49) 

to  be  taken  from  the  tables. 

The  central  angle  of  the  partial  taper  is 

9  (M2  +  Mi)  (M2  -  Mi  -  A) 


o,. 


*(50) 


or  oc  may  be  taken  from  the  tables,  being  the  difference 
between  o  for  /2  and  for  li. 


FIG.  22. 


The  common  radial  line,  OF,  divides  this  angle,  oc, 
proportionally  to  the  degrees  of  curve  of  the  two  branches 
of  the  compound  curve. 

*  See  footnote,  page  28. 


30  THE   RAILROAD  TAPER 


Let  DA  be  tangent  to  the  curve  of  lesser  degree  at  the 
beginning  of  the  partial  taper,  and  let  the  curve  of  greater 
degree  be  backed  up  through  angle  oc,  so  that  its  tangent, 
ET,  shall  be  parallel  to  DA,  then 

tc  =  (Ri  -  R2-  dr)  sin  a,  (52) 

dc  =  Rl-R*-  tc  cot  a,  (53) 

or  =  (Ri  —  Rz)  vers  a  +  dr  cos  a.      (53a) 

From  Eqn.  4,    yc  =  tc  +  #2  sin  oc.  (54) 

From  Eqn.  5,    zc  =  dc  -+-  #2  vers  oc.  (55) 

Using  the  method  of  second  differences  with  the  tables, 
the  deflection  to  the  end  of  the  taper,  from  tangent  at  the 
beginning  of  the  partial  taper,  is 

,       (3  h  +  le  +  45)  lc  +  450  „,     . 

fc~  ~" 


+  2M2  +  A)  (M2  -  M,  -  A) 

—  fA 


] 
J. 


Equation  56  may  be  used  for  obtaining  all  the  deflec- 
tions from  tangent  at  A,  the  beginning  of  the  partial  taper, 
to  points  on  the  taper,  using  for  lc  the  distance  of  the  point 
beyond  the  beginning  of  the  partial  taper,  A,  li  remaining 
constant  from  equation  45. 

Where  MI  is  a  multiple  of  A,  the  deflections  to  points 
on  the  partial  taper  may  be  taken  directly  from  the  tables, 
transit  at  I  =  li,  deflection  to  I  =  li  +  Zc,  using  for  lc  the 
distance  of  the  point  beyond  the  beginning  of  the  partial 
taper,  A. 

*  See  footnote,  page  28. 


LOCATION  PROBLEMS  31 

The  angle  of  backsight  to  the  beginning  of  the  partial 
taper,  from  tangent  at  the  end  of  the  taper,  is 

bs  =  oc-.fc.  (58) 
The  long  chord  of  the  partial  taper  is,  from  Eqn.  2, 

lcc  =  yc  sec/c.  (59) 
20.   The  Semi- tangents  of  the  Compound  Curve. 

From  Fig.  22          A  =  c  -  a,  (60) 

h  =  g  +  a.  (61) 

By  Eqn.  8,  SD  =  ti  +  £>i  tan  J  /i  -  *  esc  /i.  (62) 

By  Eqn.  9,  DA  =  Ri  tan  \  /i  +  di  esc  Ilf  (63) 

By  Eqn.  10 

AG  =  te  +  *DC  tan  \  72  -  (dc  -  eft,)  esc  /,.  (64) 
By  Eqn.  11, 

GH  =  t2  +  D2  tan  \  I2  -  (d2  -  de)  esc  72,  (65) 

DG  =  DA  +  AG.  (66) 

Solving  triangle  DNG  gives  values  for  DN  and  AfG. 

Then  SN  =  SD  +  DN,  (67) 

NH  =  NG  +  GH.  (68) 

Where  it  is  desirable  to  obtain  the  semi-tangents  of  the 
compound  curve  without  first  finding  the  taper  functions  of 
the  partial  taper,  DG  may  be  drawn  tangent  to  MI  at  F 
and  equations  62  to  68  become  as  follows: 

SD  =  ti  +  Di  tan  i  c  -  di  esc  c,  (62a) 

DF  =  #1  tan  i  c  +  di  esc  c,  (63a) 

FG  =  fA-  tan  }  0  -  (dr  -  A)  esc  g,  (64a) 
GH  =  k  +  D2  tan  J  g  -  (4  -  *•)  esc  g,  (65a) 
DG  =  DF  +  TO,  (66a) 

>S^V  =  ^D  +  D^V,  (67) 

NH  =  NG  +  GH.  (68) 

Example.  Given  7  =  114°  24',  c  =  34°  40',  g  =  79°44', 
Mi  =  3°  00',  M-2.  =  8°  00r.  Use  Taper  1  between  tangent 

*  Dc  =  B,  +  dc. 
t  Dr  =  ft  +  dr. 


32  THE  RAILROAD  TAPER 

and  3°  00'  curve,  Taper  2  between  3°  00'  and  8°  00'  curves, 
and  Taper  2J  between  the  8°  00'  curve  and  tangent. 
Find  the  semi-tangents. 

30 
By  Eqn.  48,         lc  =  y  (8  -  3  -  1) 

=  120. 

By  Eqn.  49,  and  the  tables, 
dr  =  0.79. 

By  Eqn.  50,  oc  =  9  (8  +  3)  (8  -  3  -  1) 

=  396', 
=  6°  36'. 

9  X  3  (8  -  3  -  1) 
By  Eqn.  51a,  a  =  -          ^— 

=  108' 

=  1°  48'. 

By  Eqn.  52,  tc  =      37.47. 

By  Eqn.  53,  dc  =        1.38. 

By  Eqn.  54,  yc  =    119.80. 

By  Eqn.  55,  xc  =        6.13. 

By  Eqn.  57,  fe  =  175J'  =  2°  55|'. 

By  Eqn.  58,  6s  =  3°  40£'. 

By  Eqn.  59,  lcc  =    119.96. 

By  Eqn.  60,  /i  =  32°  52;. 

By  Eqn.  61,  /2  =  81°  32'. 

By  Eqn.  62,  SD  =    637.26. 

By  Eqn.  63,  DA  =    564.59. 

By  Eqn.  64,  AG  =    656.32. 

By  Eqn.  65,  GH  =    683.74. 

By  Eqn.  66,  DG  =  1220.91. 

FromADGW,  NG  =    727.57, 

and  DN  =1326. 06. 

By  Eqn.  67,  SN  =  1963.32. 

By  Eqn.  68,  NH  =  1411 .31. 


LOCATION  PROBLEMS 


33 


21.  Given  a  Located  Curve,  to  Compound  with  Curve 
of  Given  Radius  so  as  to  End  on  a  Given  Parallel  Tangent, 
using  Three  Tapers. 

In  Fig.  23,  given  located  curve  SAFV  of  radius  Ri, 
parallel  tangent  NH  distant  p  from  the  original  tangent 
WV,  Rz,  di,  d2)  dr,  ti  and  fe,  to  find  Ilt  72  and  NH. 


FIG.  23. 

OE  =  A  -  p  -  D2, 
OP  =  Ri-dr-  R2, 
OE 


COS       = 


OP 


=  Rj  -  R*  +  dj  -  d*  -  p 

RI  —  R%  —  dr 
c  =  I  -  g. 


*(69) 
(70) 


*  Where  the  new  tangent  is  beyond  the  original  tangent,  requir- 
ing Rz  to  be  greater  than  Kb  p  and  dr  in  equations  69,  71    and  72 


change  sign.     For  dr  see  Eqn.  49. 


34  THE  RAILROAD  TAPER 

The  new  semi-tangents  may  now  be  determined  by  the 
methods  of  the  preceding  problem.  If  the  semi-tangents 
of  the  two  parts  of  the  curve,  very  useful  in  running  in  the 
curve,  are  not  needed,  the  semi-tangents  of  the  entire 
curve  may  be  obtained  more  easily  as  follows: 

SN  =  (R!  +  di)  tan  J  /  -  p  esc  /  +  *,,  *(71) 

NH  =  (Ri  +  di)  tan  J  /  -  (Ri  -  Ri  -  dr)  sin  g 

'fe.  *f(72) 


Example.  Given  a  located  4°  30'  curve,  Taper  1J,  in- 
tersection angle  82°  14',  to  compound  onto  a  parallel 
tangent  71  ft.  beyond  the  original  tangent,  using  a  2°  45' 
curve,  Taper  1,  with  Taper  1  between  the  two  curves. 

ByEqn.69,  Cosg  = 

g  =  24°  14'. 

By  Eqn.  70,  c  =  58°  00'. 

By  Eqn.  71,  SN  =  1258.99. 

By  Eqn.  72,  NH  =  1502.68. 

22.  Given  a  Located  Curve,  to  Compound  onto  a  new 
Located  Tangent,  Point  of  Compound  Approximately 
Fixed,  full  Tapered. 

In  Fig.  24,  AN'  is  the  located  curve,  to  be  compounded 
at  about  N'  so  as  to  unite  with  located  tangent  UF. 

Run  tangent  to  located  curve  at  N',  intersecting  located 
tangent.  From  Functions  of  a  One-degree  Curve  deter- 
mine degree  of  curve,  M',  with  which  to  compound  if 
tapers  were  neglected.  Then  use  such  a  degree  of  curve 
that 

fl2  .  ff  _  *  +  *,  (73) 


and  the  resulting  g  will  be  very  nearly  g'. 

*  See  footnote,  page  33. 

t  Care  should  be  taken  to  use  a  negative  value  for  cot  /  when 
/  is  greater  than  90°. 


LOCATION  PROBLEMS  35 

OE  =  OB  +  BF  -  FE 

=  Rl  cos]gf  +  N'C  sin  g'  -  (R2  +  A), 
OP  =  Ri  -R2-dr, 

OE 
cos  g  =  Qp 

_  Ri  cos  g'  +  N'C  sin  g'  -  (ft  +  (fe)     *,-,, 
Rt-Rt-  dr 

Example.  Given  located  4°  00'  curve  ANX,  to  com- 
pound at  about  Nf  (Nf  to  be  about  the  center  of  the  taper 
between  parts  of  the  compound),  so  as  to  end  on  located 
tangent  CF.  N'C  =  340.5  ft.,  g'  =  42°  12',  both  meas- 
ured on  the  ground. 

i  c         u 


FIG.  24. 

From  Functions  of  a  1°  Curve: 
T42°12'  =  2211.0, 


R'  =  5730  X  == 


340.5 


2211 


=  882.4, 
'  =  6°29'+. 


*  For  dr  see  Eqn.  49. 


36  THE  RAILROAD  TAPER 

By  Eqn.  73,  d2  =  1 . 75  for  6°  30'  Tap.  2, 

dr  =  0.09. 

A             D                      1.75  +  0.09 
Approx.  R2  =  882 . 4 

EB 

=  881.52, 
=  881.59. 


406.53 


Use  6°  30'  curve,  Tap.  2. 


By  Eqn.  74, 


COS0   = 


550.79 
=  42°  26'. 


23.  Given  one  of  the  Semi-tangents  of  a  Compound 
Curve,  the  Intersection  Angle,  and  both  Radii,  to  find 
the  two  Central  Angles  and  the  other  Semi-tangent. 

In  Fig.  25,  given  SN,  I,  RI  and  Rz,  to  find  c,  g,  and  NH. 


FIG.  25. 

OE  =  OTF  +  WU  -  UE 

=  (Ri  +  di)  cos  /  +  TN  sin  /  -  (R2  +  4), 

f)~D   D        _    E>     J 

0# 


LOCATION  PROBLEMS  37 

(Ri  +  di)  cos  /  +  TN  sin  /  -  (fa  +  d») 

Ri-R2-  *dr 
NH  =  LU  -  LN  -  UV  +  Z2 

=  (#1  H-  di)  sin  I  -  TN  cos  7 

-  (fii  -  #2  -  *dr)  sin  0  +  fe.  f(76) 

Example.  Given  semi-tangent  £/V  =  1963.32,  MI  = 
3°  00'  Tap.  1,  M2  =  8°  00'  Tap.  2},  and  Tap.  2  between 
Mi  and  M2,  /  =  114°  24',  to  find  c,  0  and  N#. 

212.62 
By  Eqn.  75,  cos  g  =  UQ2  7g> 

g  =  79°  44'. 
By  Eqn.  76,  NH  =  1411.32. 

TAPERS  BEYOND  THE  FROG 

24.  When  the  Track  Beyond  the  Turnout  curves 
back  parallel  to  the  main  line,  there  is  seldom  room  and 
little  need  for  a  taper.  When  the  track  beyond  the  turn- 
out curves  almost  immediately  in  the  same  direction  as 
the  turnout,  the  taper  is  of  much  use,  both  as  to  looks  and 
from  practical  considerations. 

It  has  been  the  practice  on  many  roads  to  introduce  a 
short  tangent,  of  twenty  to  thirty  feet,  beyond  the  frog, 
before  beginning  the  curve.  This  answers  the  double 
purpose  of  giving  the  trackman  room  for  an  approximate 
taper,  and  by  removing  the  B.C.  from  the  heel  of  the  frog, 
prevents  the  kink  that  otherwise  is  very  likely  to  occur 
there  due  to  the  rail  joint  on  the  outside  rail.  However, 
this  presents  the  very  bad  appearance  and  gives  the  bad 
riding  of  the  carefully  avoided  "  broken-backed  curve." 
If  the  curve  begin  at  the  heel  of  the  frog  with  a  taper  of 
moderate  length,  both  appearance  and  service  are  much 
improved. 

*  For  dr  see  Eqn.  49. 

f  When  #2  is  greater  than  Ri,  dr  becomes  negative.  When  7  is 
greater  than  90°,  cos  /  becomes  negative. 


38 


THE   RAILROAD  TAPER 


Considerable  ease  in  location  will  often  be  obtained  if 
the  curve  beyond  the  taper  may  be  handled  as  a  simple 
curve  tangent  to  a  parallel  with  the  main  line,  instead  of 
with  the  tangent  through  the  frog. 

In  Fig.  26,  /  is  the  distance  from  frog  angle  point  to 
actual  point  of  frog,  usually  given  by  standard  drawings. 
It  is 

/  =  \  (g  +  v)  cot  J  F,  (77) 

where  g  =  gauge,  v  =  thickness  of  frog  point,  usually 
one-half  inch,  and  F  =  frog  angle,  m  =  length  of  frog 
from  actual  point  to  heel. 


L _f_ *u.~ w~J 

\ [p 


FIG.  26. 


Then 


p  =  HE  +  KO  -  OD 

=  (f-+m  +  t)wnF+  (R  +  d)cosF-R,     (78) 
w  =  CH  -  SH  -  CP 

=  (f  +  m  +  0  cosF  -  (R  +  d)  sinF  -/.      (79) 


LOCATION  PROBLEMS  39 

From  Eqn.  10, 
AP  =  w  +  (R+p)  tan  J  (F+I).  -  (p-d)  esc  (F+I).  (80) 

From  Eqn.  11, 
AN  =  t  +  (R+d)  tan  }  (^+7)  -  (d-p)  esc  (F+7),  (81) 


sin/ 

The  resulting  calculations  for  the  entire  problem  may 
be  longer  than  if  the  semi-tangent  BT  be  calculated  by 
Eqn.  7,  /  +  ra  added,  and  triangle  ACB  solved.  However, 
p  and  w  are  very  useful  in  a  preliminary  consideration  of 
the  location  of  the  turnout. 

Example.     Given  a  No.  9  turnout  from  a  side  track  to 
a  branch  line,  with  a  6°  00'  curve,  Tap.  2J,  intersection 
angle  72°  48',  beginning  at  the  heel  of  the  frog,    g  =  4'-8i", 
v  =  J",  m  =  8'-10",  F  =  6°  22': 
By  Eqn.  77,  /  =  42.70. 

By  Eqn.  78,  p  =  10.70  +  949.72  -  955.04 

=  5.38. 
By  Eqn.  79,  w  =  95.92  -  105.97  -  42.70 

=  -52.75. 

The  negative  value  of  w  means  that  S  lies  between  P 
andC. 

By  Eqn.  80,         AP  =  -  52.75  +  794.08  -  4.88 

=  736.45. 
By  Eqn.  81,         AN  =  44.99  +  790.  10  +  4.88 

=  839.97. 
By  Eqn.  82,         AB  =    90.45. 


CHAPTER  III 

FIELD   WORK 

SELECTION  OF  THE  TAPER 

25.  The  Selection  of  the  Proper  Taper  for  use  on 
each  curve  is  an  important  function  of  the  engineer.  As 
suggested  elsewhere,  the  proper  length  of  taper  to  use  is 
one  on  which  the  super-elevation  of  rails  given  the  main 
curve  may  be  run  off  on  the  taper  at  from  one-half  to  one 
inch  per  rail  length.  This  will  suggest  the  use  of  Taper  1, 
1J  or  2,  for  curves  that  do  not  limit  the  speed  of  trains, 
with  a  maximum  length  of  about  360  ft.,  and  thence 
gradually  decreasing  in  length  as  velocity  and  super- 
elevation decrease,  to  a  minimum  of  90  ft.  Although 
some  very  short  tapers  are  given  in  the  tables,  they  are 
for  exceptional  conditions.  The  Table  of  Comparative 
Values  will  aid  in  the  selection  of  the  taper. 

A  table  of  the  theoretical  elevation  needed  to  counter- 
act the  centrifugal  forces  is  included  with  the  other  tables, 
copied  from  the  Proceedings  of  the  American  Railway 
Engineering  Association,  but  as  the  super-elevation  de- 
pends more  upon  the  velocity  of  trains  than  upon  the 
degree  of  curve,  and  as  this  velocity  varies  with  trains, 
seasons  and  traffic,  an  arbitrary  rule,  usually  a  compromise, 
is  in  force  on  most  roads. 

The  beginning  and  end  of  every  taper  should  be  clearly 
marked  for  the  track  foreman,  so  that  he  may  know 
between  what  points  he  is  to  give  the  rails  a  gradually 
increasing  super-elevation.  The  proper  maintenance  of 
this  gradual  increase,  however,  is  no  easy  task,  even  when 
the  points  are  well  marked,  and  if  the  distance  over  which 

40 


FIELD  WORK  41 

it  occurs  be  too  great,  and  the  increase  in  super-elevation 
be  too  gradual,  it  is  very  likely  to  become  and  remain  very 
irregular,  totally  destroying  the  value  of  the  taper. 
Therefore,  though  theoretically  a  curve  tapered  through- 
out might  seem  perfection,  as  maintained  practically  it 
would  be  a  very  poorly  riding  curve.  Too  long  a  taper 
is  as  bad  as  none. 

The  somewhat  prevalent  practice  of  fitting  a  circular 
curve  to  the  ground,  and  tangents  at  some  undetermined 
offset  therefrom,  then  connecting  the  two  by  the  neces- 
sary taper  to  suit .  the  offset,  has  the  support  of  a  few 
eminent  engineers,  but  should  be  condemned  as  slipshod 
work.  It  is  rarely  the  case  that  more  careful  thought 
and  investigation  will  fail  to  place  some  other  curve  or 
tangents  in  a  different  location,  with  a  suitable  taper 
which  should  be  chosen  for  totally  different  reasons. 
The  weight  and  velocity  of  trains,  the  weight  and  char- 
acter of  the  engines  and  cars  whose  direction  is  to  be 
changed,  and  the  amount  and  quickness  of  the  change, 
should  alone  govern  the  choice  of  taper. 

If  any  additional  cost  of  construction  be  involved  in 
the  proper  choice  of  taper,  which  is  doubtful  with  good 
choice  of  location,  it  would  be  justified  by  the  resulting 
lessened  operating  costs.  Most  certainly  the  higher 
quality  of  locating  ability  required  to  properly  place  a 
line  to  suit  economic  conditions  is  well  able  to  solve  this 
lesser  problem. 

LAYING  OUT  A  TAPERED  CURVE 

26.  After  the  location  of  tangents  and  main  curve  has 
been  decided  upon  and  choice  of  taper  made,  the  B.C. 
and  E.C.  should  be  set  by  measurement  from  the  P.I. 
The  taper  at  each  end  should  then  be  run  from  the  B.C. 
and  E.G.,  the  C.C.  to  the  main  curve  being  set  preferably 
by  long  chord  from  the  B.C.  and  E.G.,  and  then  the  main 
curve  should  be  run  from  C.C.  to  C.C.  of  tapers,  or  from 


42  THE  RAILROAD  TAPER 

each  C.C.  to  the  middle  of  the  curve  if  it  is  unusually 
long. 

It  is  usual,  especially  for  final  track  centers,  to  stake  the 
taper  every  thirty  feet  from  B.C.  and  E.G.,  closing  to  the 
C.C.  with  a  short  chord  where  the  length  of  taper,  I,  is 
not  a  multiple  of  30  ft. ;  but  on  location,  and  especially  for 
slope  staking  on  construction,  it  is  often  desirable  to 
stake  the  station  and  plus  fifty  points,  which  may  be  done 
by  obtaining  the  deflections  from  Eqn.  107. 

If  the  main  curve  is  of  moderate  length,  it  is  usual, 
with  the  transit  at  the  C.C.,  to  turn  the  transit  at  the 
B.C.  with  the  backsight  deflection  set  on  the  reverse  side  of 
the  vernier,  thus  bringing  the  instrument  into  tangent 
at  the  C.C.  with  the  vernier  at  zero,  and  thence  run  in 
the  main  curve  with  foresight  on  the  other  C.C.  If  the 
main  curve  is  unusually  long,  and  the  other  C.C.  not 
visible,  this  involves  running  a  comparatively  long  curve 
from  a  short  backsight,  frequently  resulting  in  poor 
closing.  In  this  case  a  flag  may  be  set  at  an  offset  from 
the  initial  tangent  equal  to  x  of  the  taper,  preferably 
opposite  the  P.I.,  but  in  any  event  as  far  from  the  C.C. 
as  the  curve  to  be  run  from  it.  With  the  vernier  set  at 
the  total  central  angle  of  the  taper,  o,  and  the  instrument 
directed  at  this  flag,  bringing  it  into  a  direction  parallel 
to  the  initial  tangent,  the  instrument  will  then  be  tangent 
to  the  taper  at  the  C.C.  when  the  vernier  is  brought  to 
zero. 

PASSING  OBSTRUCTIONS  ON  THE  TAPER 

27.   Curve  run  from  Main  Curve  Produced. 

It  frequently  happens  that  either  the  B.C.  or  C.C.  of 
the  taper  is  inaccessible,  when  it  is  often  necessary  to  run 
the  main  curve  from  the  offset  B.C.  of  a  simple  curve,  or 
from  a  point  on  the  simple  curve  produced  backward  as 
far  as  necessary  beyond  the  B.C.  of  the  simple  curve,  such 
point  being  set  by  offset  from  the  initial  tangent. 


FIELD  WORK 


43 


In  Fig.  27,  point  C  is  the  B.C.  of  the  simple  curve,  D 
the  B.C.  and  B  the  C.C.  of  the  taper.  The  central  angle 
of  the  portion  of  the  simple  curve  from  C  to  B  is  the 
central  angle  of  the  taper,  o.  Point  A,  a  distance  back 
on  the  simple  curve  equal  to  CB,  and  about  opposite  the 
B.C.  of  the  taper,  is  a  point  frequently  chosen  from  which 


FIG.  27. 

to  run  the  main  curve,  because  of  the  ease  with  which  its 
position  may  be  determined.  Its  offset  from  tangent  is 
x  of  the  taper,  and  its  distance  along  the  tangent  from  the 
B.C.  of  the  taper  is 

n  =  y  —  2  R  sin  o. 

From  Eqn.  4,      R  sin  o  =  y  —  t, 

then  n  =  2t-y.  (83) 

Any  other  point,  E,  may  be  chosen,  its  offset  from 
tangent  being, 

r=d  +  R  vers  I',  (84) 

where  I'  is  the  central  angle  of  the  curve  back  of  the  B.C. 
of  the  simple  curve. 

The  distance  along  the  tangent  from  the  B.C.  of  the 
taper  is 

s  =  Z- fivers/'.  (85) 

This  is  merely  equivalent  to  suggesting  that  the  main 
curve  may  be  run  in  beyond  the  C.C.  from  any  point  back 
of  the  C.C.  by  considering  it  a  simple  curve  from  a  parallel 
tangent. 


44  THE  RAILROAD  TAPER 

28.   Curve  run  from  a  Point  Beyond  the  C.C. 

When  on  location  the  C.C.  is  in  a  bad  position  to  occupy 
with  the  transit,  a  point  may  be  set  on  the  main  curve 
beyond  the  C.C.  from  a  point  on  the  semi-tangent,  by 
working  the  problem  under  paragraph  17  backwards. 
The  plus  of  the  point  to  be  set  is  assumed,  and  from  that 
angle  a,  Fig.  28,  is  known.  D  and  A  B  are  known.  Then 

tann=^,  (86) 

OB  =  —  ,  (87) 

cosn 

g  =  a  -  (90°  -  n) 
=  a  +  n  -  90°.  (88) 

Solving  triangle  OBP,  two  sides  and  the  included  angle 
given, 

tan  (c  +  }(/)=  §|^|  tan  \  g,  (89) 

b  =  180°  -  n  -  c,  (90) 


sine 
For  point  P',  equations  88  and  90  become 

g'  =  90°  -  (a'  +  n),  (88a) 

&'  =  180°  -  n  +  c',  (90a) 

the  other  equations  remaining  the  same. 

Example.  Given  a  4°  00'  curve,  Taper  1|,  C.C.  in  a 
poor  location  for  a  transit  set  up,  the  second  station 
beyond,  137.9  ft.  from  the  C.C.,  being  a  particularly 
good  transit  point.  On  the  same  ridge  with  this  second 
station  is  a  point  on  the  semi-tangent,  PoST,  260.89  ft. 
from  the  B.C.  Determine  the  angle  and  distance  from 
PoST  to  PoC. 

a  =  4  X  0.6'  X  137.9  +  2°  36' 
=  8°  07'. 

By  Eqn.  86,  n  =  82°  13'. 

By  Eqn.  87,  OB  =  1446.50. 


By  Eqn.  88, 
By  Eqn.  89, 

By  Eqn.  90, 
By  Eqn.  91, 


FIELD  WOKK 
0  =  0°  20'. 

\- 1  g  =  30°  50', 
c  =  30°  40'. 
6  =  67°  07'. 

BP  =  16.34. 


45 


FIG.  28. 


FIG.  29. 


29.  Taper  Points  set  by  Offset. 

When  both  the  B.C.  and  the  C.C.  are  inconvenient  to 
occupy  with  the  transit,  the  taper  points  may  be  set  by 
offset  from  the  tangent,  using  x  and  y,  or  they  may  be 
set  by  offset  from  the  tangent  on  the  first  half  and  from 
the  curve  on  the  second  half.  In  the  latter  case  the  offset 
from  the  curve  is  the  same  as  from  tangent  at  equal 
distances  from  B.C.  and  C.C. 

30.  Taper  Points  set  from  Point  on  Semi-tangent. 

A  few  or  all  of  the  taper  points  are  occasionally  set  by 
measurement  and  angle  from  a  point  on  the  semi-tangent, 
PoST,  by  a  solution  of  the  triangle  in  Fig.  29. 


tan  b  = 


AB  = 


CB-y1 

CB-y 

cos  b 


(92) 
(93) 


46  .  THE   RAILROAD  TAPER 

31.   Curve  Notes  for  the  Field  Book. 

Assuming  the  B.C.  at  Sta.  489  +  15.2,  the  transit 
notes  for  a  tapered  curve  would  be  as  follows: 

497 

O  +62.2  4°41i'    2°  37*'     E.G. 

+32.2  4°  14i' 

498  +02.2  3°41i' 

+  72.2  3°  02|' 

+42.2  2°  17J' 

495  +12.2  l°26i' 

+82.2  0°29i'  I  =  195 

0  +67.2  13°23J'  13°  23*'     C.C.  Tap.  2  o  =  7°  18|' 

494  10°  52' 

+50  8°  59*'    •  TA  =  41°  24' 

493  7°  07'  T  =  387.16 

+50     .  5°  14*'  Ext.  =    55.64 

492  3\22'  R  =  764.08 

+50  1°29*'  Lm  =  357.0 

O    491  +10.2  2°  37*'    4e41i'     C.C.  7°  30'  R          A  =  26°  46*' 

+  95.2  2°  16*' 

+65.2  1°39'  t  =    97.41 

+  35.2  1°07*'  d  =      2.71 

490  +05.2  0°42'  x  =      8.93 

+  75.2  0°22*'  lc  =  194.86 

+45.2  0°09'  I  =  195 

©  +15.2  0°  00'  B.C.  Tap.  2  o  =  7°  18|' 

489 

In  the  above,  the  deflections  for  the  taper  at  the  E.G. 
are  given  from  tangent  at  the  C.C.,  calculated  from  the 
equation 


,,  _ 

J  300 

A 

an  empirical  equation  derived  from  the  tables  by  the 
method  of  second  differences,  in  which  I  is  the  distance 
from  B.C.  or  E.G.  to  the  point  for  which  deflection  is 
wanted,  and  s  is  the  distance  of  the  transit  from  B.C.  or 
E.G.  The  equation  gives  the  deflection  in  minutes,  and 
is  from  tangent  at  the  transit.  The  deflections  from 
tangent  at  the  C.C.  given  in  the  notes  above  are  merely 
to  make  the  notes  complete.  Usually  both  tapers  are 
run  from  B.C.  and  E.G.  towards  the  main  curve,  using 
the  same  set  of  deflections. 


CHAPTER  IV 
OFFICE  WORK 

32.  Plotting  a  Tapered  Curve. 

In  the  plotting  of  tapered  curves,  it  is  customary,  after 
plotting  the  tangents  and  locating  the  B.C.  and  E.G.,  to 
draw  the  simple  curve  in  its  offset  position,  then  draw  a 
curve  of  radius  twice  that  of  the  main  curve  extending 
from  B.C.  and  E.G.  a  distance  equal  to  I,  the  length  of 
the  taper,  coinciding  with  the  main  curve  at  the  C.C. 
The  difference  between  such  a  curve  and  the  exact  taper 
is  noticeable  only  if  the  drawing  be  on  a  very  large  scale. 

33.  Paper  Location  with  Tapers. 

The  usual  method  of  paper  location  with  tapers,  as 
indeed  frequently  in  location  on  the  ground,  is  to  offset 
the  simple  curve  from  its  tangents  an  amount  equal  to 
d  for  the  taper  chosen.  If  curves  predominate,  they  will 
be  located  first,  and  with  parallel  curves  near  their  prob- 
able ends  of  radius  greater  by  d,  the  tangents  may  be 
drawn  in  their  offset  position.  If  tangents  predominate, 
they  will  be  drawn  first,  with  short  parallel  tangents  near 
their  probable  ends,  and  the  curves  drawn  tangent  to 
these  offset  tangents. 

After  the  location  of  the  simple  offset  curves,  and  the 
tangents,  the  tapers  may  be  drawn  or  merely  indicated  as 
desired,  depending  upon  the  scale  of  map  used. 

34.  Curve  Compensation  with  Tapers. 

It  is  a  practice  on  some  roads  to  use  a  compensation  on 
the  taper  of  half  that  on  the  main  curve.  An  exactly 
equivalent  result,  with  a  simpler  and  more  satisfactory 
grade  line,  is  obtained  if  the  change  of  grade  from  tangent 

47 


48  THE   RAILROAD  TAPER 

to  curve  be  made  at  the  point  opposite  the  offset  position 
of  the  B.C.  of  the  simple  curve,  which  is  at  the  middle  of 
the  taper,  and  a  suitable  vertical  curve  inserted. 

This  brings  correct  theoretical  and  practical  results, 
increasing  the  compensation  in  exact  ratio  with  the  curva- 
ture, and  is  well  suited  to  location  in  rough  country,  where 
final  consideration  of  the  taper  must  be  postponed  until 
the  location  of  curves  and  tangents. 

35.   Right  of  Way  Descriptions  with  Tapered  Curves. 

A  suitable  description  of  right  of  way  when  the  taper 
is  used  would  be:  ".  .  .  ,  the  center  line  of  said  rail- 
road extending  thence  along  a  tangent  N  1°  14'  W  to 
station  4937  -f-  48.2,  a  distance  of  465  ft.,  thence  along 
a  taper  -to  the  right  of  curvature  increasing  one  degree 
every  thirty  feet  to  station  4938  +  98.2,  a  distance  of 
150  ft.,  thence  along  a  6°  00'  curve  to  the  right  (radius 
955.04)  to  station  4945  +  42.8,  a  distance  of  644.6  ft., 
thence  along  a  taper  to  the  right  of  curvature  decreasing 
one  degree  every  thirty  feet  to  station  4946  +  92.8,  a 
distance  of  150  ft.,  thence  along  a  tangent  N  46°  26|'  E  to 
station  5052  +  47.1,  a  distance  of  554.3  ft.,  .  .  .  ." 


CHAPTER  V 
TAPERING  OLD  TRACK 

36.  The  Tapering  of  Old  Track,  located  and  con- 
structed with  circular  curves  only,  calls  for  most  careful 
analysis  of  the  conditions  encountered,  and  generally 
repays  in  results  obtained  a  great  deal  of  otherwise  need- 
less transit  work.  Although  after  long  experience  in  this 
particular  work  one  may  decide  almost  offhand  upon  the 
taper  to  use  and  the  changes  to  make  in  tangents  and 
curves,  tjie  beginner  must  usually  make  more  than  one 
trial  before  arriving  at  the  best  results  obtainable,  which 
usually  combines  with  the  most  satisfactory  curve  for 
operation  the  lowest  cost  in  making  the  change  in  align- 
ment. Such  work  is  frequently  complicated  by  bridges 
on  the  curves  or  tangents  near  the  curves,  limiting  the 
amount  track  may  be  thrown,  or  by  the  slow,  though 
frequently  very  decided,'  sliding  of  embankment  on  side 
hill  work,  distorting  the  original  circular  curve. 

It  will  also  frequently  be  found  that  the  trackman  has 
thrown  the  B.C.  in  towards  the  center  of  curvature, 
lining  in  an  approximate  taper,  and  if  the  curve  be  short, 
may  have  taken  care  of  the  additional  curvature  by  throw- 
ing the  middle  of  the  curve  out.  If  the  curve  be  long,  it 
will  probably  be  found  that  a  short  portion  beyond  this 
approximate  taper  is  of  unduly  heavy  curvature.  On  the 
other  hand,  the  passage  of  heavy  engines  of  rigid  wheel 
base  has  frequently  thrown  the  curve  out  at  the  B.C.  and 
E.C.,  with  which  is  coupled  unduly  heavy  curvature  just 
beyond.  When  allowed  sufficient  time,  the  engineer  will 
rerun  the  original  simple  curve  to  determine  just  where 
the  track  is,  before  attempting  to  insert  tapers.  The 

49 


50  THE  RAILROAD  TAPER 

resulting  transit  notes,  with  measured  offsets  to  the  track 
as  it  is  found,  together  with  notes  showing  all  bridges  and 
openings,  narrow  embankments  and  cuts,  and  points 
where  embankments  and  cuts  may  be  cheaply  widened 
and  where  not,  will  give  the  engineer  the  data  for  a  care- 
ful office  study  of  the  conditions  to  be  encountered. 
With  an  understanding  of  the  various  methods  suggested 
below  of  inserting  tapers,  a  tentative  plan  may  be  decided 
upon  for  each  curve,  subject  to  revision  when  running  in 
the  curves.  Since  the  method  involving  the  lowest  maxi- 
mum shift  of  track  is  the  shifting  of  both  curve  and  tan- 
gents, which  will  then  influence  the  next  curve  in  each 
direction,  the  entire  series  of  curves  and  tangents  must  be 
studied  as  a  whole. 

The  insertion  of  tapers  in  old  circular  track  will  gener- 
ally be  accompanied  by  a  resurfacing  out  of  face  of  the 
entire  portion  of  the  line  involved,  so  that  usually  there 
is  no  need  to  confine  the  shifting  of  track  to  the  curves, 
if  the  shifting  of  tangents  also  will  improve  results.  If 
the  track  may  be  ballasted  at  the  same  time  the  cost  of 
inserting  tapers  is  practically  eliminated. 

37.  Although,  unfortunately,  in  the  insertion  of  tapers 
in  old  circular  curve  alignment,  the  physical  conditions 
encountered  must  very  largely  influence  the  choice  of 
taper,  frequently  necessitating  the  use  oT  short  tapers, 
the  operating  conditions  due  to  amount  and  direction  of 
grade,  velocity,  weight  and  character  of  trains,  amount 
of  traffic,  direction  of  heaviest  tonnage,  etc.,  must  not  be 
lost  sight  of,  but  must,  as  largely  as  possible,  influence  the 
choice  of  taper. 

METHODS  OF  INSERTING  TAPERS  IN  CIRCULAR 
TRACK.     SIMPLE  CURVES 

38.  Shifting  the  Curve.     The  simplest,  though  usually 
the  least  desirable,  method  of  making  room  for  the  taper 
is  by  shifting  the  curve  radially  inward,  the  degree  of 


TAPERING  OLD  TRACK 


51 


curve  remaining  unchanged.  The  amount  track  will  be 
shifted  will  vary  from  zero  at  the  new  B.C.  and  E.G.,  to 
MN,  Fig.  30,  at  the  middle  of  the  curve  (with  equal  tapers 


at  the  two  ends)  ,  varying  approximately  with  the  distance 
from  B.C.  or  E.G. 

MN  =  00'  =  AB  =  d  sec  }  L  (95) 

As  this  method  will  shorten  the  line  somewhat,  by  an 
amount  equal  to 

S  =  2£ 


necessitating  the  cutting  of  rails  under  usual  conditions, 
it  is  to  be  avoided  if  possible  except  at  summits,  where 
the  rail  joints  are  likely  to  be  quite  open,  or  at  sags, 
where  the  cutting  of  rails  would  be  desirable  to  relieve 
joints  already  too  tight. 

This  method  is  limited  to  small  values  of  d  and  7,  to 
avoid  throwing  the  inner  rail  onto  or  close  to  fresh  em- 
bankment, accentuating  the  tendency  of  the  inner  rail  to 
settle,  especially  on  single  and  double  track  roads,  where 
the  super-elevation  must  fit  the  speed  of  the  fastest  trains, 
throwing  a  large  part  of  the  weight  of  slower  trains  upon 
the  inner  rail. 


52 


THE   RAILROAD  TAPER 


39.  Shifting  the  Tangents.  Where  consecutive  curves 
are  of  contrary  flexure,  and  in  a  lesser  degree  where  of 
similar  flexure,  shifting  a  short  tangent,  Fig.  31,  forms  a 
desirable  method  of  making  room  for  the  taper.  The 
tangent  is  shifted  at  its  ends  amounts  approximately  equal 
to  d  of  the  tapers  adopted. 


b 


FIG.  31. 

With  this  method  it  is  usual  first  to  fit  the  main  portion 
of  each  curve  as  closely  as  desired  by  trial,  not  over- 
looking provision  for  inserting  a  proper  taper  at  the  other 
end  of  each  curve,  then  connecting  the  two  curves  by  a 
meander  line,  preferably  the  old  tangent,  and  proceeding  as 
in  paragraph  15.  Due  to  irregularities  found  in  the  track 
it  is  not  likely  that  these  curves,  fitted  to  the  track  as 
found,  will  be  exactly  tangent  to  the  old  tangent,  but  this 
is  immaterial. 

With  this  method  the  length  of  track  is  slightly  in- 
creased, but  seldom  enough  to  be  objectionable. 


TAPERING   OLD  TRACK 


53 


40.  Shifting  Curve  and  Tangents.  This  method, 
Fig.  32,  usually  gives  the  most  desirable  rearrangement  of 
alignment,  involving  the  'lowest  maximum  throw  of  track, 
and  maintaining  the  length  of  line  practically  unchanged. 
To  obtain  this  latter  result  the  tangents  should  be  thrown 
out  about  two-thirds  of  d  of  the  taper  opposite  the  old 
B.C.  and  E.G.,  then  run  to  an  intersection  and  the  curve 
run  in.  The  new  curve  will  cross  the  old  one  on  the 


FIG.  32. 

second  half  of  the  taper,  the  entire  central  portion  of  the 
new  curve  lying  inside  the  old  curve,  about  one-third  d 
from  it  at  the  middle  of  the  curve.  These  amounts  will 
vary  with  the  length  of  the  tangents  and  their  resulting 
angular  shift. 

41.  Increasing  the  Degree  of  Curve.  Where  the  pre- 
ceding methods  involve  too  great  a  throw  of  track  with 
the  taper  desired,  the  curve  may  be  sharpened,  as  in  Fig. 
33,  enough  to  make  room  for  the  taper. 

d  =  (R  -  R')  vers  \  I.  (97) 

As  the  value  of  d  depends  upon  both  R'  and  the  taper 
chosen,  which  latter  itself  is  influenced  by  Rf,  trial  solu- 
tions of  the  second  half  of  this  equation  must  be  made 


54 


THE   RAILROAD  TAPER 


until  a  desirable  value  is  obtained  for  d.  For  a  desirable 
simplification  of  this  point  see  the  solution  in  the  next 
problem. 


With  this  method  of  inserting  tapers,  track  will  be 
thrown  in  toward  the  center  of  curvature  throughout  the 
curve,  the  throw  being  a  maximum  about  at  the  C.C.  of 
the  tapers.  This  will  be 

h  =  (R-  R')  vers  (}  7  -  o).  (98) 

42.  Increasing  Degree  and  Shifting  Curve.  Curves 
of  odd  minutes  curvature  are  to  be  avoided  as  much  as 
possible,  as,  the  trackman  being  unfamiliar  with  them, 
they  are  difficult  to  maintain.  If  the  curve  of  the  pre- 
ceding problem  be  sharpened  so  that  (R  —  R')  vers  \  I 
will  give  a  value  greater  than  d,  and  the  curve  shifted  away 
from  the  center  of  curvature  to  bring  it  tangent  to  the 
original  tangents,  Fig.  34,  a  desirable  realignment  is 
obtained. 

At  the  middle  of  the  curve  track  will  now  be  thrown  out 
an  amount 


or 


h  =  [(R-  R')  vers  }  I  -  d]  sec  J 7,  (99) 

h=   (R-Df)  exsec  }  7  -  d.  (99a) 


TAPERING   OLD  TRACK 


55 


For  preliminary  study,  where  I  is  not  large,  the  factor 
sec  \  I  may  be  disregarded  as  insignificant.  The  new 
curve  will  now  lie  within  the  old  throughout  the  tapers 


and  a  portion  of  the  main  curve,  and  outside  the  old  curve 
on  the  middle  portion  of  the  main  curve,  a  very  desirable 
rearrangement.  The  best  balance  is  obtained  by  making 
h  equal  about  one-third  d,  when  the  maximum  throw  of 
track  at  the  three  points  of  greatest  divergence  will  be 
about  equal. 

43.  Maintaining  Length  of  Line  Unchanged.  With 
the  proper  value  of  h,  Eqn.  99,  R'  may  be  assumed  such  as 
to  leave  the  length  of  line  unchanged. 

The  length  of  line  from  A  to  B,  Fig.  34,  by  the  new 
alignment  is 

(100) 


The  length  of  line  from  A  to  B  by  the  old  alignment  is 
1007 


L  =  2AC 


M 


1007 
M 


(101) 


56  THE   RAILROAD  TAPER 

The  table  on  page  115  gives  the  values  of  M',  I  and  h  for 
a  large  range  of  M  and  /  with  the  more  important  tapers. 
Where  curves  are  sharper  than  ten-degree,  tangents  will 
be  short,  and  paragraphs  39  and  40  will  give  better  results. 
Where  I  is  greater  than  given  in  the  tables,  paragraph  46 
will  usually  give  better  results  unless  curves  of  fractional 
minutes  curvature  are  used. 

When  the  value  of  M  or  7  lies  between  those  given  in 
the  table,  M'  may  be  interpolated  and  then  checked  by 
equations  100  and  101,  recalculating  when  necessary.  If 
on  the  first  trial  L'  —  L  is  too  great,  decrease  M1  '.  The 
irregularity  of  h  in  the  table  is  due  to  using  the  nearest 
minute  for  M'. 

Equations  100  and  101  are  based  upon  fifty-foot  chord 
measurement  of  both  old  and  new  curves,  but  any  dis- 
crepancy over  arc  measurement  introduced  is  small  and 
may  usually  be  neglected.  With  both  curves  based  upon 
a  fifty-foot  chord,  increasing  a  5°  00'  curve  to  a  5°  15' 
curve  introduces  a  discrepancy  due  to  arc  excess  of  only 
0.0002  ft.  per  hundred  feet.  With  the  old  curve  based 
upon  a  hundred-foot  chord  and  the  new  curve  based  upon 
a  fifty-foot  chord  the  difference  in  arc  excess  is  quite  large, 
especially  with  the  higher  degrees  of  curve,  but  this  is 
almost  exactly  balanced  by  the  decreased  value  of  AC, 
equation  101,  over  that  assumed  for  the  table,  due  to  the 
increased  value  of  R. 

For  arc  measurement  equations  100  and  101  become 

L'(arc)=Z  +  ^,  (lOOa) 

L(arc)  =  2*-2(fl-£')tani/ 


Using  these  equations,  if  a  10°  00'  curve,  I  =  50°  00', 
be  sharpened  to  a  10°  17'  curve,  Tap.  3,  as  given  in  the 
table,  the  new  alignment  would  be  0.04  ft.  longer  than  the 
old.  The  curve  would  have  to  be  sharpened  to  a  10°  16' 
curve,  Tap.  3,  to  maintain  the  length  unchanged. 


TAPERING   OLD  TRACK 


57 


The  length  of  line  may  be  maintained  unchanged  with 
many  of  the  other  problems  in  this  chapter  by  arranging 
by  the  manner  of  shifting  curves  and  tangents  that 
approximately  half  the  new  alignment  lie  within  and  half 
without  the  old. 

44.  Increasing  Degree  and  Shifting  Curve  and  Tan- 
gents* Where  /  is  large  the  smallest  increase  of  curvature 
Avoiding  odd  minutes  will  frequently  give  too  large  a  value 


for  h  of  equation  99,  when  to  avoid  throwing  the  curve  out 
too  much  at  the  middle  the  tangents  may  be  thrown  in 
at  the  B.C.  and  C.C.  of  the  old  curve,  as  in  Fig.  35. 

c  +  h  =  (R-R')  vers \l  -  d  (approx.).         (102) 

For  the  best  balance  of  track  h  should  about  equal  c. 
The  new  tangents  may  be  run  to  an  intersection,  new 
external  calculated  and  h  checked  on  the  ground,  and  if 
satisfactory  the  curve  run  in. 

45.  Any  of  the  preceding  methods  may  be  combined 
in  any  one  curve,  by  shifting  the  tangent  at  one  end  only 


58 


THE   RAILROAD  TAPER 


FlG.  38. 


TAPERING   OLD  TRACK  59 

either  the  full  or  partial  amount  of  d,  and  the  curve  at  the 
other  end,  or  the  tangents  may  be  shifted  different  amounts 
at  the  two  ends  of  the  curve,  thus  taking  advantage  of 
different  widths  of  roadbed  available  at  different  points. 
On  heavy  grades  a  shorter  taper  may  be  used  at  the 
lower  end  of  the  curve  when  room  is  limited,  due  to  the 
lower  speed  of  ascending  trains. 

46.  Increasing  Degree  of  Portion  of  Curve.  Where  I 
is  too  great  to  sharpen  the  entire  curve,  and  the  tangents 
too  long  to  be  shifted  advantageously,  a  portion  only  of 
the  curve  may  be  sharpened  to  make  room  for  the  taper, 
as  in  Fig.  36.  The  difference  between  the  degree  of  the 
old  curve  and  the  portion  sharpened  should  preferably  not 
be  greater  than  the  curvature  increment  of  the  taper  to 
be  used,  and  may  often  be  less. 

-r  (103) 


/'  must  of  course  be  larger  than  the  central  angle  of  the 
taper,  or  the  increase  in  curvature  to  provide  the  offset 
has  been  too  great.  I'  should  not  be  less  than  twice  0, 
and  may  usually  be  as  much  as  three  times  o. 

COMPOUND  CURVES 

47.  Any  combination  of    the   preceding   problems  of 
this  chapter  may  be  used  in  inserting  tapers  at  the  ends 
or  between  the  branches  of  old  compound  curves.     A  few 
of  the  possible  combinations  follow. 

48.  Shifting  the  Curves.     In  Fig.  37  the  relative  values 
of  the  various  d's  are  exaggerated,  increasing  the  angular 
change  in  the  position  of  the  point  of  compounding.     How- 
ever, this  arrangement  disturbs  the  branch  of  sharper 
curvature  disproportionately,  which  may  be  modified  as  in 
Fig.  38  by  shifting  the  branch  of  lighter  curvature  for  the 
entire  amount  of  dr. 


60 


THE   RAILROAD  TAPER 


49.   Shifting  Curves  and  Tangents.     Where  both  the 
curves  and  tangents  may  be  shifted,  as  in  Fig.  39,  the 


FIG.  39. 


maximum  throw  of  track  may  be  much  reduced.     This 
aids  particularly  with  the  branch  of  sharper  curvature. 

50.   Shifting  the  Lighter  Branch,  and  Increasing  the 
Curvature  of  the  Sharper  Branch.     This  change  of  align- 


FIG.  40. 


ment  is  shown  in  Fig.  40.  With  this  may  of  course 
be  combined  the  shifting  of  the  end  of  the  sharper  branch 
to  avoid  curvature  of  odd  minutes,  as  in  paragraph  42,  or 
also  the  shifting  of  one  or  both  tangents  as  in  paragraph  44. 


TAPERING   OLD  TRACK 


FIELD  WORK 


61 


51.  The  principal  objection  to  compound  curve  align- 
ment is  the  difficulty  trackmen  find  in  maintaining  it  to 
proper  curvature.  It  follows  that  in  inserting  tapers  in 
compound  curves  it  is  seldom  profitable  to  make  any 
lengthy  calculations  based  upon  the  theoretical  location 
of  the  curves.  They  will  be  somewhere  else. 

The  best,  method  of  procedure  is  to  pick  out  portions  of 
the  curves  of  apparently  fairly  uniform  curvature,  which 
can  be  tested  with  the  mid-ordinate  to  a  hundred-foot 
tape  held  against  the  rail  at  the  ends  of  the  tape,  set  up  the 
transit  on  these  portions,  extend  them  as  far  in  each  direc- 
tion as  they  fit  the  track  fairly  well,  then  turn  tangents  and 
intersect  as  in  Fig.  41.  It  will  usually  be  found  that  these 


FIG.  41. 

curves,  fitted  to  the  track,  are  not  tangent  to  each  other  nor 
to  the  original  tangents,  the  offsets  from  which  may  or  may 
not  be  in  a  direction  to  facilitate  the  insertion  of  tapers. 


62  THE   RAILROAD  TAPER 

At  the  tangent  ends,  the  existing  offset  from  tangent  is, 
from  Fig.  41, 

c  =  AB  sin  a  —  R  vers  a.  (104) 

Between  the  two  branches  of  the  compound  curve,  the 
existing  offset  along  the  common  radial  line  may  be  found 
from  Fig.  41. 

A  solution  of  right  triangles  OPK  and  O'PS  will  give  the 
sides  O'P  and  OP  and  the  included  angle  /  of  triangle 
O'PO,  from  the  solution  of  which  00'  and  angle  g  may  be 
obtained,  from  which  angle  n  may  be  obtained.  Then 

m  =  b  —  n,  (105) 

e  =  R'  -  R  -  O'O.  (106) 

With  these  data  the  shifting  of  curves  and  tangents 
most  desirable  may  be  decided  upon.  'Then  to  insure 
perfect  fitting  of  the  curves  to  each  other  and  to  their 
tangents,  the  tangents  may  be  located  in  their  new  posi- 
tions, the  longer  of  the  two  branches  located  in  its  new 
position  by  the  problem  in  paragraph  12,  and  then  the 
shorter  of  the  two  branches  located  by  the  problem  in 
paragraph  22. 


TABLES 


63 


RADII 

25 


,                 sin  — 

, 

Deg. 

Radius. 

Log  rad. 

Deg. 

Radius. 

Log.  rad. 

1 
Deg. 

Radius. 

Log.  rad. 

0°00' 

Infinite 

1°00' 

5729.60 

3.75812 

2°  00' 

2864.82 

3.45710 

1 

343775.76 

•5.53627 

1 

5635.67 

3.75095 

1 

2841.15 

3.45349 

2 

171887.41 

5.23524 

2 

5544.77 

3.74388 

2 

2817.86 

3.44992 

3 

114591.60 

5.05915 

3 

5456.76 

3.73693 

3 

2794.95 

3.44637 

4 

85943.67 

4.93421 

4 

5371.50 

3.73010 

4 

2772.41 

3.44286 

5 

68754.94 

4.83730 

5 

5288.86 

3.72336 

5 

2750.23 

3.43937 

6 

57295.78 

4.75812 

6 

5208.73 

3.71673 

6 

2728.41 

3.43591 

7 

49110.67 

4.69118 

7 

5130.99 

3.71020 

7 

2706.93 

3.43248 

8 

42971.83 

4.63318 

8 

5055.53 

3.70377 

8 

2685.78 

3.42907 

9 

37197.19 

4.58203 

9 

4982.26 

3.69743 

9 

2664.96 

3.42569 

10 

34377.47 

4.53627 

10 

4911.09 

3.69118 

10 

2644.46 

3.42234 

11 

31252.24 

4.49488 

11 

4841.92 

3.68502 

11 

2624.27 

3.41901 

12 

28647.89 

4.45709 

12 

4774.67 

3.67895 

12 

2604.39 

3.41571 

13 

26444.80 

4.42234 

13 

4709.26 

3.67295 

13 

2584.81 

3.41243 

14 

24555.34 

4.39016 

14 

4645.63 

3.66704 

14 

2565.52 

3.40918 

15 

22918.30 

4.36018 

15 

4583.69 

3.66121 

15 

2546.52 

3.40595 

16 

21485.92 

4.33215 

16 

4523.37 

3.65546 

16 

2527.79 

3.40274 

17 

20222.04 

4.30583 

17 

4464.63 

3.64979 

17 

2509.35 

3.39956 

18 

19098.60 

4.28100 

18 

4407.39 

3.64418 

18 

2491.16 

3.39640 

19 

18093.80 

4.25753 

19 

4351.60 

3.63865 

19 

2473.24 

3.39327 

20 

17188.74 

4.23524 

20 

4297.21 

3.63319 

20 

2455.57 

3.39015 

21 

16370.20 

4.21405 

21 

4244.16 

3.62779 

21 

2438.16 

3.38706 

22 

15626.13 

4.19385 

22 

4192.40 

3.62246 

22 

2420.99 

3.38399 

23 

14946.70 

4.17555 

23 

4141.87 

3.61720 

23 

2404.06 

3.38095 

24 

14323.89 

4.15606 

24 

4092.58 

3.61200 

24 

2387.36 

3.37792 

25 

13750.99 

4.13833 

25 

4044.43 

3.60686 

25 

2370.90 

3.37491 

26 

13222.11 

4.12130 

26 

3997.39 

3.60177 

26 

2354.67 

3.37193 

27 

12732.40 

4.10491 

27 

3951.46 

3.59676 

27 

2338.65 

3.  ,36896 

28 

12277.67 

4.08912 

28 

3906.56 

3.59179 

28 

2322.85 

3.36602 

29 

11854.34 

4.07388 

29 

3862.66 

3.58689 

29 

2307.26 

3.36310 

30 

11459.17 

4.05915 

30 

3819.75 

3.58203 

30 

2291.88 

3.36019 

31 

11089.52 

4.04491 

31 

3777.77 

3.57724 

31 

2276.70 

3.35731 

32 

10742.97 

4.03112 

32 

3736.71 

3.57249 

32 

2261  .  72 

3.35444 

33 

10417.66 

4.01777 

33 

3696.53 

3.56779 

33 

2246.94 

3.35159 

34 

10111.03 

4.00480 

34 

3657.21 

3.56315 

34 

2232.35 

3.34876 

35 

9822.15 

3.99221 

35 

3618.71 

3.55855 

35 

2217.95 

3.34595 

36 

9549.31 

3.97997 

36 

3581.02 

3.55401 

36 

2203.73 

3.34316 

37 

9291.65 

3.96809 

37 

3544.10 

3.54951 

37 

2189.70 

3.34038 

38 

9046.71 

3.95649 

38 

3507.93 

3.54505 

38 

2175.84 

3.33763 

39 

8814.75 

3.94521 

39 

3472.50 

3.54064 

39 

2162.15 

3.33489 

40 

8594.38 

3.93421 

40 

3437.78 

3.53628 

40 

2148.64 

3.33216 

41 

8384.76 

3.92341 

41 

3403.74 

3.53196 

41 

2135.29 

3.32946 

42 

8185.12 

3.91303 

42 

3370.37 

3.52768 

42 

2122.12 

3.32677 

43 

7994.77 

3.90281 

43 

3337.65 

3.52344 

43 

2109.09 

3.32410 

44 

7813.13 

3.89282 

44 

3305.56 

3.51924 

44 

2096.24 

3.32144 

45 

7639.45 

3.88306 

45 

3274.08 

3.51509 

45 

2083.53 

3.31880 

46 

7473.38 

3.87352 

46 

3243.19 

3.51097 

46 

2070.98 

3.31618 

47 

7314.37 

3.86418 

47 

3212.88 

3.50689 

47 

2058.56 

3.31357 

48 

7161.99 

3.85503 

48 

3183.13 

3,50285 

48 

2046.33 

3.31098 

49 

7015.82 

3.84608 

49 

3153.93 

3.49885 

40 

2034.22 

3.30840 

50 

6875.51 

3.83730 

50 

3125.26 

3.49489 

50 

2022.26 

3.30584 

51 

6740.70 

3.82870 

51 

3097.10 

3.49096 

51 

2010.43 

3.30329 

52 

6611.09 

3.82027 

52 

3069.45 

3.48706 

52 

1998.74 

3.30076 

53 

6486.33 

3.81200 

53 

3042.29 

3.48320 

53 

1987.19 

3.29824 

54 

6366.22 

3.80388 

54 

3015.60 

3.47937 

54 

1975.77 

3.29574 

55 

6250.46 

3.79591 

55 

2989.38 

3.47558 

55 

1964.48 

3.29325 

56 

6138.85 

3.78809 

56 

2963.61 

3.47182 

56 

1953.32 

3.29077 

57 

6031  .  15 

3.78640 

57 

2938.28 

3.46809 

57 

1942.28 

3.28831 

58 

5927.17 

3.77285' 

58 

2913.38 

3.46440 

58 

1931.37 

3.28587 

59 

5826.71 

3.76542 

59 

2888.89 

3.46073 

59 

1920.58 

3.28343 

64 


THE   RAILROAD  TAPER 


RADII  —  (Continued) 


Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

3°  00' 

1909.91 

3.28101 

4°  00' 

1432.47 

3.15608 

5°  00' 

1146.01 

3.05919 

1 

1899.36 

3.27861 

1 

1426.52 

3.15430 

1 

1142.20 

3.05774 

2 

1888.93 

3.27622 

2 

1420.63 

3.15248 

2 

1138.42 

3.05630 

3 

1878.61 

3.27384 

3 

1414.78 

3.15069 

3 

1134.66 

3.05487 

4 

1868.40 

3.27147 

4 

1408.99 

3.14891 

4 

1130.93 

3.05344 

5 

1858.30 

3.26912 

5 

1403.24 

3.14713 

5 

1127.20 

3.05201 

6 

1848.31 

3.26677 

6 

1397.53 

3.14536 

6 

1123.54 

3.05059 

7 

1838.42 

3.26445 

7 

1391.88 

3.14360 

7 

1119.88 

3.04917 

8 

1828.65 

3.26213 

8 

1386.26 

3.14185 

8 

1116.24 

3.04776 

9 

1818.97 

3.25983 

9 

1380.70 

3.14010 

9 

1112.63 

3.04635 

10 

1809.40 

3.25753 

10 

1375.17 

3.13836 

10 

1109.04 

3.04495 

11 

1799.93 

3.25525 

11 

1369.70 

3.13662 

11 

1105.48 

3.04355 

12 

1790.56 

3.25299 

12 

1364.26 

3.13490 

12 

1101.94 

3.04216 

13 

1781.27 

3.25073 

13 

1358.87 

3.13318 

13 

1098.42 

3.04077 

14 

1772.09 

3.24849 

14 

1353.53 

3.13146 

14 

1094.92 

3.03938 

15 

1763.01 

3.24625 

15 

1348.21 

3.12976 

15 

1091.44 

3.03800 

16 

1754.01 

3.24403 

16 

1342.95 

3.12806 

16 

1087.99 

3.03662 

17 

1745.11 

3.24182 

17 

1337.72 

3.12637 

17 

1084.56 

3.03525 

18 

1736.30 

3.23962 

18 

1332.54 

3.12468 

18 

1081.15 

3.03389 

19 

1727.57 

3.23744 

19 

1327.39 

3.12300 

19 

1077.76 

3.03252 

20 

1718.93 

3.23526 

20 

1322.29 

3.12133 

20 

1074.39 

3.03116 

21 

1710.38 

3.23309 

21 

1317.22 

3.11966 

21 

1071.05 

3.02981 

22 

1701.92 

3.23094 

22 

1312.19 

3.11800 

22 

1067.72 

3.02846 

23 

1693.53 

3.22879 

23 

1307.21 

3.11634 

23 

1064.42 

3.02711 

24 

1685.23 

3.22666 

24 

1302.26 

3.11470 

24 

1061.13 

3.02577 

25 

1677.01 

3.22454 

25 

1297.34 

3.11305 

25 

1057.87 

3.02443 

26 

1668.87 

3.22242 

26 

1292.47 

3.11142 

26 

1054.62 

3.02310 

27 

1660.81 

3.22032 

27 

1287.63 

3.10979 

27 

1051.40 

3.02177 

28 

1652.83 

3.21823 

28 

1282.82 

3.10817 

28 

1048.19 

3.02044 

23 

1644.92 

3  21614 

29 

1278.05 

3.10655 

29 

1045.01 

3.01912 

30 

1637.09 

3.21407 

30 

1273.32 

3.10494 

30 

1041.84 

3.01780 

31 

1629.33 

3.21201 

31 

1268.62 

3.10333 

31 

1038.69 

3.01649 

32 

1621.64 

3.20996 

32 

1263.96 

3.10173 

32 

1035.57 

3.01518 

33 

1614.03 

3.20791 

33 

1259.33 

3.10014 

33 

1032.46 

3.01387 

34 

1606.49 

3.20588 

34 

1254.73 

3.09855 

34 

1029.37 

3.01257 

35 

1599.02 

3.20385 

35 

1250.17 

3.09697 

35 

1026.29 

3.01127 

36 

1591.62 

3.20184 

36 

1245.64 

3.09539 

36 

1023.24 

3.00998 

37 

1584.28 

3.19983 

37 

1241.15 

3.09382 

37 

1020.20 

3.00869 

38 

1577.01 

3.19784 

38 

1236.68 

3.09226 

38 

1017.19 

3.00740 

39 

1569.81 

3.19585 

39 

1232.25 

3.09070 

39 

1014.19 

3.00612 

40 

1562.68 

3.19387 

40 

1227.85 

3.08915 

40 

1011.20 

3.00484 

41 

1555.61 

3.19190 

41 

1223.48 

3.08760 

41 

1008.24 

3.00356 

42 

1548.60 

3.18994 

42 

1219.14 

3.08606 

42 

1005.29 

3.00229 

43 

1541.66 

3.18799 

43 

1214.84 

3.08452 

43 

1002.36 

3.00102 

44 

1534.78 

3.18605 

1   44 

1210.56 

3.08299 

44 

999.45 

2.99976 

45 

1527.96 

3.18411 

45 

1206.31 

3.08146 

45 

996.55 

2.99850 

46 

1521.19 

3.18218 

46 

1202.10 

3.07994 

46 

993.67 

2.99724 

47 

1514.49 

3.18027 

47 

1197.91 

3.07842 

47 

990.81 

2.99599 

48 

1507.85 

3.17736 

48 

1193.75 

3.07691 

48 

987.96 

2.99474 

49 

1501.27 

3.17646 

49 

1189.62 

3.07541 

49 

985.13 

2.99350 

50 

1494.74 

3.17457 

50 

1185.51 

3.07391 

50 

982.32 

2.99225 

51 

1488.27 

3.17268 

51 

1181.44 

3.07241 

51 

979.52 

2.99101 

52 

1481.86 

3.17081 

52 

1177.40 

3.07092 

52 

976.74 

2.98978 

53 

1475.50 

3.16894 

53 

1173.38 

3.06944 

53 

973.97 

2.98855 

54 

1469.19 

3.16708 

54 

1169.39 

3.06796 

54 

971.22 

2.98732 

55 

1462.94 

3.16523 

55 

1165.43 

3.06649 

55 

968.48 

2.98609 

56 

1456.75 

3.16338 

56 

1161.49 

3.06502 

56 

965.76 

2.98487 

57 

1450.60 

3.16155 

57 

1157.58 

3.06355 

57 

963.06 

2.98365 

58 

1444.50 

3.15972 

58 

1153.70 

3.06209 

58 

960.37 

2.98244 

59 

1438.46 

3.15790 

59 

1149.84   3.06064 

59 

957.70 

2.98123 

TABLES 


65 


RADII  —  (Continued) 


Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

6°  00' 

955.04 

2.98002 

7°  00' 

818.64 

2.91309 

8°  00' 

716.34 

2.85512 

952.39 

2.97882 

1 

816.69 

2.91206 

1 

714.85 

2.85422 

2 

949.76 

2.97762 

2 

814.76 

2.91103 

2 

713.37 

2.85332 

3 

947.15 

'  2.97642 

3 

812.83 

2.91000 

3 

711.90 

2.85242 

4 

944.55 

2.97522 

4 

810.92 

2.90898 

4 

710.43 

2.85152 

5 

941.96 

2.97403 

5 

809.01 

2.90795 

5 

708.96 

2.85062 

6 

939.39 

2.97284 

6 

807.11 

2.90693 

6 

707.50 

2.84973 

7 

936.83 

2.97166 

7 

805.22 

2.90592 

7 

706.05 

2.84884 

8 

934.28  • 

2.97048 

8 

803.34 

2.90490 

8 

704.60 

2.84795 

9 

931.75 

2.96930 

9 

801.47 

2.90389 

9 

703.16 

2.84706 

10 

929.23 

2.96812 

10 

799.61 

2.90288 

10 

701.73 

2.84617 

11 

926.73 

2.96695 

11 

797.75 

2.90187 

11 

700.30 

2.84528 

12 

924.24 

2.96578 

12 

795.91 

2.90086 

12 

698.88 

2.84440 

13 

921.76 

2.96462 

13 

794.07 

2.89986 

13 

697.46 

2.84352 

14 

919.29 

2.96346 

14 

792.24 

2.89886 

14 

696.05 

2.84264 

15 

916.84 

2.96230 

15 

790.42 

2.89786 

15 

694.64 

2.84176 

16 

914.41 

2.96114 

16 

788.61 

2.89686 

16 

693.24 

2.84089 

17 

911.98 

.2.95999 

17 

786.80 

2.89587 

17 

691.85 

2.84t)01 

18 

909.57 

2.95884 

18 

785.01 

2.89487 

18 

690.46 

2.83914 

19 

907.17 

2.95769 

19 

783.22 

2.89388 

19 

689.08 

2.83827 

20 

904.78 

2.95655 

20 

781.44 

2.89290 

20 

687.70 

2.83740 

21 

902.41 

2.95540 

21 

779.67 

2.89191 

21 

686.33 

2.83653 

22 

900.05 

2.95427 

22 

777.91 

2.89093 

22 

684.96 

2.83567 

23 

897.70 

2.95313 

23 

776.15 

2.88995 

23 

683.60 

2.83480 

24 

895.36 

2.95200 

24 

774.40 

2.88897 

24 

682.25 

2.83394 

25 

893.04 

2.95087 

25 

772.66 

2.88799 

25 

680.90 

2.83308 

26 

890.72 

2.94974 

26 

770.93 

2.88702 

26 

679.55 

2.83222 

27 

888.42 

2.94862 

27 

769.21 

2.88604 

27 

678.21 

2.83136 

28 

886.13 

2.94750 

28 

767.49 

2.88507 

28 

676.88 

2.83051 

29 

883.86 

2.94638 

29 

765.78 

2.88410 

29 

675.55 

2.82966 

30 

881.59 

2.94527 

30 

764.08 

2.88314 

30 

674.22 

2.82880 

31 

879.34 

2.94416 

31 

762.39 

2.88218 

31 

672.90 

2.82795 

32 

877.09 

2.94305 

32 

760.70 

2.88121 

32 

671.59 

2.82710 

33 

874.86 

2.94194 

33 

759.02 

2.88025 

33 

670.28 

2.82626 

34 

872.64 

2.94084 

34 

757.35 

2.87930 

34 

668.98 

2.82541 

35 

870.44 

2.93974 

35 

755.69 

2.87834 

35 

667.68 

2.82457 

36 

868.24 

2.93864 

36 

754.03 

2.87739 

36 

666.39 

2.88373 

37 

866.05 

2.93754 

37 

752.38 

2.87644 

37 

665.10 

2  82289 

38 

863.88 

2.93645 

38 

750.74 

2.87549 

38 

663.81 

2^82205 

39 

861.71 

2.93536 

39 

749.10 

2.87454 

39 

662.54 

2.82121 

40 

859.56 

2.93428 

40 

747.48 

2.87360 

40 

661.26 

2.82037 

41 

857.42 

2.93319 

41 

745.85 

2.87265 

41 

659.99 

2.81954 

42 

855.28 

2.93211 

42 

744.24 

2.87171 

42 

658.73 

2.81871 

43 

853.16 

2.93103 

43 

742.63 

2.87077 

43 

657.47 

2.81788 

44 

851.05 

2.92996 

44 

741.03 

2.86984 

44 

656.22 

2.81705 

45 

848.95 

2.92888 

45 

739.44 

2.86890 

45 

654.97 

2.81622 

46 

846.86 

•  2.92781 

46 

737.86 

2.86797 

46 

653.72 

2.81539 

47 

844.78 

2.92674 

47 

736.28 

2.86704 

47 

652.48 

2.81457 

48 

842.71 

2.92568 

48 

734.70 

2.86611 

48 

651.25 

2.81375 

49 

840.65 

2.92461 

49 

733.14 

2.86519 

49 

650.02 

2.81293 

50 

838.60 

2.92355 

50 

731.58 

2.86426 

50 

648.79 

2.81211 

51 

836.56 

2.92250 

51 

730.03 

2.86334 

31 

647.57 

2.81129 

52 

834.53 

2.92144 

52 

728.48 

2.86242 

52 

646.35 

2.81047 

53 

832.51 

2.92039 

53 

726.94 

2.86150 

53 

645.14 

2.80966 

54 

830.50 

2.91934 

54 

725.41 

2.86058 

54 

643.93 

2.80884 

55 

828.50 

2.91829 

55 

723.88 

2.85967 

55 

642.73 

2.80803 

56 

826.51 

2.91725 

56 

722.36 

2.85875 

56 

641.53 

2.80722 

57 

824.53 

2.91620 

57 

720.85 

2.85784 

57 

640.34 

2.80641 

58 

822.55 

2:91516 

58 

719.34 

2.85693 

58 

639.15 

2.80560 

59 

820.59 

2.91413 

59 

717.84 

2.85603 

59 

637.96 

2.80480 

66 


THE  RAILROAD  TAPER 


RADII  —  (Continued) 


Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

9°  00' 

636.78 

2.80399 

10°  00' 

573.14 

2.75826 

11°  00' 

521.07 

2.71690 

1 

635.60 

2.80319 

1 

572.19 

2.75754 

1 

520.28 

2.71624 

2 

634.43 

2.80239 

2 

571.24 

2.75682 

2 

519.50 

2.71558 

3 

633.27 

2.80159 

3 

570.29 

2.75610 

3 

518.71 

2.71493 

4 

632.10 

2.80080 

4 

569.35 

2.75538 

4 

517.93 

2.71427 

5 

630.94 

2.79999 

5 

568.41 

2.75466 

5 

517.16 

2.71362 

6 

629.79 

2.79920 

6 

567.47 

2.75394 

6 

516.38 

2.71297 

7 

628.64 

2.79840 

7 

566.53 

2.75323 

7 

515.61 

2.71232 

8 

627.49 

2.79761 

8 

565.60 

2.75251 

8 

514.84 

2.71167 

9 

626.35 

2.79682 

9 

564.67 

2.75180 

9 

514.07 

2.71102 

10 

625.21 

2.79603 

10 

563.75 

2.75109 

10 

513.30 

2.71037 

11 

624.08 

2.79524 

11 

562.83 

2.75038 

11 

512.54 

2.70972 

12 

622.95 

2.79445 

12 

561  91 

2.74967 

12 

511.77 

2.70908 

13 

621.82 

2.79367 

13 

560.99 

2.74896 

13 

511.02 

2.70843 

14 

620.70 

2.79288 

14 

560.08 

2.74825 

14 

510.26 

2  70779 

15 

619.58 

2.79210 

15 

559.17 

2.74754 

15 

509.50 

2.70714 

16 

618.47 

2.79132 

16 

558.26 

2.74684 

16 

508.75 

2.70650 

17 

617.36 

2.79054 

17 

557.36 

2.74613 

17 

508.00 

2.70586 

18 

616.25 

2.78976 

18 

556.46 

2.74543 

18 

507.25 

2.70522 

19 

615.15 

2.78898 

19 

555.56 

2.74473 

19 

506.50 

2.70458 

20 

614.05 

2.78821 

20 

554.66 

2.74403 

20 

505.76 

2.70394 

21 

612.96 

2.78743 

21 

553.77 

2.74333 

21 

505.02 

2.70330 

22 

611.87 

2.78666 

22 

552.88 

2.74263 

22 

504.28 

2.70267 

23 

610.78 

2.78589 

23 

551.99 

2.74193 

23 

503.54 

2.70203 

24 

609.70 

2.78512 

24 

551.11 

2.74124 

24 

502.80 

2.70140 

25 

608.62 

2.78435 

25 

550.23 

2.74054 

25 

502.06 

2.70076 

26   607.55 

2.78358 

26 

549.35 

2.73985 

26 

501.33 

2.70013 

27   606.48 

2.78281 

27 

548.47 

2.73916 

27 

500.60 

2.69950 

28   605.41 

2.78205 

28 

547.60 

2.73847 

28 

499.88 

2.69887 

29   604.35 

2.78129 

29 

546.73 

2.73777 

29 

499.16 

2.69824 

30 

603.29 

2.78052 

30 

545.87 

2.73709 

30 

498.43 

2.69761 

31 

602.23 

2.77976 

31 

545.00 

2.73640 

31 

497.72 

2.69698 

32 

601  .  18 

2.77900 

32 

544.14 

2.73571 

32 

497.00 

2.69635 

33 

600.13 

2.77824 

33 

543.28 

2.73502 

33 

496.28 

2.69572 

34 

599.08 

2.77749 

34 

542.42 

2.73434 

34 

495.56 

2.69510 

35 

598.04 

2.77673 

35 

541.57 

2.73365 

35 

494.85 

2.69447 

36 

597.00 

2.77598 

36 

540.72 

2.73297 

36 

494.14 

2.69385 

37 

595.97 

2.77523 

37 

539.87 

2.73229 

37 

493.43 

2.69323 

38* 

594.93 

2.77447 

38 

539.03 

2.73161 

38 

492.73 

2.69261 

39 

593.91 

2.77372 

39 

538.18 

2.73093 

39 

492.02 

2.69198 

40 

592.89 

2.77297 

40 

537.34 

2.73025 

40 

491.32 

2.69136 

41 

591.87 

2.77223 

41 

536.51 

2.72957 

41 

490.62 

2.69074 

42 

590.85 

2.77148 

42 

535.67 

2.72890 

42 

489.92 

2.69013 

43 

589.84 

2.77074 

43 

534.84 

2.72822 

43 

489.22   2.68951 

44 

588.83 

2.76997 

44 

534.01 

2.72755 

44 

488.53  j  2.68889 

45 

587.83 

2.76925 

45 

533.18 

2.72687 

45 

487.84 

2.68828 

46 

586.82 

2.76851 

46 

532.35 

2.72620 

46 

487.15 

2.68766 

47 

585.82 

2.76777 

47 

531.53 

2.72553 

47 

486.46 

2.68705 

48 

584.83 

'  2.76703 

48 

530.71 

2.72486 

48 

485.77 

2.68643 

49 

583.84 

2.76629 

49 

529.90 

2.72419 

49 

485.09 

2.68582 

50 

582.85 

2.76556 

50 

529.08 

2.72352 

50 

484.40 

2.68521 

51 

581.86 

2.76482 

51 

528.27 

2.72286 

51 

483.72 

2.68460 

52 

580.88 

2.76409 

52 

527.46 

2.72219 

52 

483.04 

2.68400 

53 

579.90 

2.76335 

53 

526.65 

2.72152 

53 

482.37 

2.68338 

54 

578.93 

2.76262 

54 

525.85 

2.72086 

54 

481  .  69 

2.68277 

55 

577.95 

2.76189 

55 

525.05 

2.72020 

55 

481.02 

2.68216 

56 

576.98 

2.76116 

56 

524.25 

2.71953 

56 

480.35 

2.68156 

57 

576.02 

2.76044 

57 

523.45 

2.71887 

57 

479.68 

2.68095 

58 

575.05 

2.75971 

58 

522.65 

2.71821 

58 

479.01 

2.68035 

59 

574.10 

2.75899 

59 

521.86 

2.71755 

59 

478.35 

2.67974 

TABLES 


67 


RADII  —  (Continued) 


Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

12°  00' 

477.68 

2.67914 

13°  00' 

440.97 

2.64441 

14°  00' 

409.51 

2.61226 

1 

477.02 

2.67854 

1 

440.41 

2.64386 

1 

409.02 

2.61175 

2 

476.36 

2.67794 

2 

439.85 

2.64330 

2 

408.54 

2.61123 

3 

475.70 

2.67734 

3 

439.29 

2.64275 

3 

408.05 

2.61072 

4 

475.05 

2.67674 

-  4 

438.73 

2.64219 

4 

407.57 

2.61020 

5 

474.39 

2.67B14 

5 

438.17 

2.64164 

5 

407.09 

2.60969 

6 

473.74 

2.67554 

6 

437.61 

2.64109 

6 

406.61 

2.60918 

7 

473.09 

2.67494 

7 

437.05 

2.64054 

7 

406.13 

2.60867 

8 

472.44 

2.67435 

8 

436.50 

2.63999 

8 

405.66 

2.60816 

9 

471.79 

2.67375 

9 

435.95 

2.63944 

9 

405.18 

2.60764 

10 

471.15 

2.  .67316 

10 

435.40 

2.63889 

10 

404.70 

2.60713 

11 

470.50 

2.67256 

11 

434.85 

2.63834 

11 

404.22 

2.60662 

12 

469.86 

2.67197 

12 

434.30 

2.63779 

12 

403.75 

2.60611 

13 

469.22 

2.67138 

13 

433.75 

2.63724 

13 

403.28 

2.60560 

14 

468.58 

2.67078 

14 

433.21 

2.63669 

14   402.81 

2.60510 

15 

467.94 

2.67019 

15 

432.66 

2.63615 

15   402.33 

2.60459 

16 

467.31 

2.66960 

16 

432.12 

2.63560 

16   401.87 

2.60408 

17 

466.68 

2.66901 

17 

431.58 

2.63506 

17  i  401.40 

2.60357 

18 

466.04 

2.66843 

18 

431.04 

2.63451 

18 

400.93 

2.60307 

19 

465.41 

2.66784 

19 

430.50 

2.63397 

19 

400.46 

2.60256 

20 

464.79 

2.66725 

20 

429.96 

2.63343 

20 

400.00 

2.60206 

21 

464.16 

2.66667 

21 

429.42 

2.63289 

21 

399.53 

2.60155 

22 

463.53 

2.66608 

22 

428.89 

2.63235 

22 

399.07 

2.60105 

23 

462.91 

2.66550 

23 

428.36 

2.63181 

23 

398.61 

2.60055 

24 

462.29 

2.66491 

24 

427.82 

2.63127 

24 

398.15 

2.60005 

25 

461.67 

2.66433 

25 

427.29 

2.63073 

25 

397.69  i  2.59954 

26 

461.05 

2.66375 

26 

426.76 

2.63019 

26 

397.23 

2.59904 

27 

460.43 

2.66317 

27 

426.23 

2.62965 

27 

396.77 

2.59854 

28 

459.82 

2.66259 

28 

425.71 

2.62911 

28 

396.32 

2.59804 

29 

459.20 

2.66201 

29 

425.18 

2.62858 

29 

395.86 

2.59754 

30 

458.59 

2.66143 

30 

424.66 

2.62804 

30 

395.41 

2.59704 

31 

457  98 

2.66085 

31 

424.14 

2.62750 

31 

394  95 

2.5C655 

32 

457.37 

2.66027 

32 

423.62 

2.62697 

32 

394.50 

2.59605 

33 

456.77 

2.65970 

33 

423.09 

2.62644 

33 

394.05 

2  59555 

34 

456.16 

2.65912 

34 

422.57 

2.62590 

34 

393  .  60 

2.59505 

35 

455.56 

2.65855 

35 

422.06 

2.62537 

35 

393.15 

2  59456 

36 

454.96  |  2.65797 

36 

421.54 

2.62484 

36 

3S2.70 

2.59406 

37 

454.36 

2.65740 

37 

421.02 

2.62431 

37 

392.56 

2.59357 

38 

453  .  76 

2.65682 

38 

420.51 

2.62378 

38 

391.81 

2.59307 

39 

453.16 

2.65625 

39 

420.00 

2.62325 

39 

391.36 

2.59258 

40 

452.57 

2.65568 

40 

419.49 

2.62272 

40 

390.91 

2.59209 

41 

451.97 

2.65511 

41 

418.98 

2.62219 

41 

390.48 

2.59160 

42 

451.38 

2.65454 

42 

418.47 

2.62166 

42 

390.04 

2.59110 

43 

450.79 

2.65397 

43 

417.96 

2.62133 

43 

389.60 

2.59061 

44 

450.20 

2.65340 

44 

417.45 

2.62061 

44 

389.16 

2.59012 

45 

449.61 

2.65284 

45 

416.95 

2.62008 

45 

388.72 

2.58963 

46 

449.02 

2.65227 

46 

416.44 

2.61955 

46 

388.28 

2.58914 

47 

448.44 

2.65170 

47 

415.94 

2.61903 

47 

387.84 

2.58865 

48 

447.86 

2.65114 

48 

415.44 

2.61851 

48 

387.40 

2.58816 

49 

447.28 

2.65057 

49 

414.94 

2.61798 

49 

386.97 

2.58767 

50 

446.70 

2.65001 

50 

414.44 

2.61746 

50 

386.53 

2.58719 

51 

446.12 

2.64945 

51 

413.94 

2.61694 

5*1 

386  .  10 

2.58670 

52 

445.54 

2.64889 

52 

413.44 

2.61642 

52 

385.67 

2.58621 

53 

444.96 

2.64832 

53 

412.95 

2.61589 

53 

385.24 

2.58573 

54 

444.39 

2.64776 

54 

412.45 

2.61537 

54 

384.81 

2.58524 

55 

443.82 

2.64720 

55 

411.96 

2.61485 

55 

384  '.38 

2  58476 

56 

443.25 

2.64664 

56 

411.47 

2.61434 

56 

383.95 

2.58423 

57 

442.67 

2.64608 

57 

410.98 

2.61382 

57 

383.52 

2  58379 

58 

442.11 

2.64553 

58 

410.49 

2.61330 

58 

383.09 

2.58331 

59 

441.54 

2.64497 

59 

410.00 

2.61278 

59 

382.67 

2.58282 

68 


THE   RAILROAD  TAPER 


RADII  —  (Continued) 


Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

15°  00' 

382.25 

2.58234 

19°  00' 

301.91 

2.47988 

23°  00' 

249.53 

2.39712 

05 

380.14 

2.57994 

05 

300.59 

2.47797 

15 

246.86 

2.39245 

10 

378.05 

2.57755 

10 

299.28 

2.47608 

30 

244.24 

2.38782 

15 

375.99 

2.57518 

15 

297.99 

2.47420 

45 

241.68 

2.38324 

20 

373.95 

2.57281 

20 

296.71 

2.47233 

25 

371.93 

2.57046 

25 

295.44 

2.47047 

24°  00' 

239.17 

2.37871 

30 

369.93 

2.56812 

30 

294.18 

2.46861 

15 

236.71 

2.37422 

35 

367.96 

2.56580 

35 

292.94 

2.46678 

30 

234.30 

2.36978 

40 

366.00 

2.56348 

40 

291.69 

2.46492 

45 

231.95 

2.36539 

45 

364.07 

2.56118 

45 

290.46 

2.46309 

50 

362.16 

2.55890 

50 

289.25 

2.46127 

25°  00' 

229.64 

2.36104 

55 

360.26 

2.55662 

55 

288.04 

2.45945 

15 

227.37 

2.35673 

30 

225.15 

2.35248 

16°  00' 

358.39 

2.55436 

20°  00' 

286.84 

2.45764 

45 

222.98 

2.34827 

05 

356.54 

2.55211 

05 

285.66 

2.45585 

10 

354.70 

2.54986 

10 

284.48 

2.45405 

26°  00' 

220.84 

2.34408 

15 

352.88 

2.54763 

15 

283.31 

2.45226 

15 

218.75 

2.33995 

20 

351.09 

2.54542 

20 

282.15 

2.45048 

30 

216.69 

2.33584 

25 

349.31 

2.54321 

25 

281.00 

2.44871 

45 

214.68 

2.33179 

30 

347.55 

2.54101 

30 

279.86 

2.44695 

35 

345.80 

2.53883 

35 

278.73 

2.44518 

27°  00' 

212.70 

2.32776 

40 

344.08 

2.53666 

40 

277.61 

2.44344 

15 

210.76 

2.32379 

45 

342.37 

2.53450 

45 

276.50 

2.44170 

30 

208.85 

2.31983 

50 

340.68 

2.53234. 

50 

275.40 

2.43996 

45 

206.98 

2.31593 

55 

339.00 

2.53020 

55 

274.30 

2.43823 

28°  00' 

205.14 

2.31205 

17°  00' 

337.34 

2.52807 

21°  (xy 

273.22 

2.43651 

15 

203.33 

2.30820 

05 

335.70 

2.52595 

05 

272.14 

2.43479 

30 

201.56 

2.30440 

10 

334.07 

2.52384 

10 

271.07 

2.43309 

45 

199.81 

2.30062 

15 

332.46 

2.52174 

15 

270.01 

2.43138 

20 

330.87 

2.51965 

20 

268.96 

2.42969 

29°  00' 

198.10 

2.29688 

25 

329.29 

2.51758 

25 

267.91 

2.42799 

15 

196.42 

2.29318 

30 

327.72 

2.51551 

30 

266.88 

2.42632 

30 

194.76 

2.28950 

35 

326.17 

2.51344 

35 

265.86 

2.42465 

45 

193.13 

2.28585 

40 

324.64 

2.51140 

40 

264.83 

2.42298 

30°  00' 

191.53 

2.28224 

45 

323.12 

2.50936 

45 

263.82 

2.42131 

50 

321.61 

2.50733 

50 

262.83 

2.41966 

55 

320.12 

2.50531 

55 

261.82 

2.41800 

18°  OO7 

318.64 

2.50330 

22°  00' 

260.83 

2.41637 

05 

317.17 

2.50129 

05 

259.85 

2.41472 

10 

315.72 

2.49930 

10 

258.88 

2.41310 

15 

314.28 

2.49732 

15 

257.91 

2.41147 

20 

312.86 

2.49534 

20 

256.95 

2.40986 

25 

311.44 

2.49337 

25 

256.00 

2.40824 

30 

310.04 

2.49142 

30 

255.05 

2.40664 

35 

308.66 

2.48948 

35 

254.12 

2.40504 

40 

307.28 

2.48754 

40 

253.19 

2.40344 

45 

305.92 

2.48561 

45 

252  .  26 

2.40185 

50 

304.57 

2.48368 

50 

251.35 

2.40027 

55 

303.23 

2.48177 

55 

250.44 

2.39870 

TABLES  69 

NOTATION 
« 

A  =  Curvature  increment  of  taper  per  30  ft.  of  taper. 
M  =  Degree  of  main  circular  curve. 
I  —  Length  of  taper  from  the  B.C.  of  the  taper. 
/  =  Deflection  from  tangent  at  the  B.C.  of  the  taper. 
ft  /i>  /2>  etc.  =•  Deflection  from  tangent  at  some  point  on  the 

taper. 
6  =  Deflection  from  tangent  at  the  end  of  the  taper,  C.C.  of 

main  curve,  to  the  B.C.  of  the  taper. 
o  =  Central  angle  of  the  taper  at  each  end  of  the  curve. 
d  =  Circular  curve  offset  from  tangent  to  provide  room  for 

the  taper. 
t  =  Distance   along   the   tangent,    from   opposite   the   offset 

position  of  the  B.C.  of  the  simple  curve  to  the  B.C.  of 

the  taper. 

D  =  R  +  d,  where  R  is  the  radius  of  the  main  circular  curve. 
x  and  y  are  co-ordinates  of  points  on  the  taper,  referred  to  the 

tangent  at  the  B.C.  of  the  taper  as  meridian,  and  to  a 

radial  line  at  the  B.C.  of  the  taper  as  base. 
Ic  =  Long  chord  of  the  taper,  from  the  B.C.  of  the  Taper  to 

points  on  the  taper. 
5  =  Distance  along  the  taper,  from  the  B.C.  of  the  taper  to  the 

transit,  when  an  intermediate  set-up  is  necessary. 
T  =  Semi-tangent  of  the  entire  tapered  curve,  from  P.I.  to 

B.C.  or  E.G. 
E  =  External,  the  distance  from  the  P.I.  to  a  point  on  the 

curve  radially  opposite. 

EQUATIONS 

Eqn.  No. 

T  =  t  +  D  tan  |  7.  (7)  Both  ends,  the  same 

taper. 
Tt  =  t  +  D  tan  £7  -  d  esc  7.  (8)  Tapered  end  —  other 

end  not  tapered. 
Tn  =  R  tan  \  I  +  d  esc  7.  (9)  Untapered  end — other 

end  Xapered. 
T1  =  ti  +  Di  tan  \  I  -  (di  -  dj  esc  7.    (10)  Both     ends     tapered 

with  different  tapers. 
T-2  =  t-i  +  D2  tan  £  /  -  (£  -  dj  esc  7.    (11)  Both     ends    tapered, 

with  different  tapers. 
E  =  D  sec  \  I  -  R.  (12)  Both   ends  the  same 

taper. 


70  THE  EAILROAD  TAPER 

EMPIRICAL  EQUATIONS  DERIVED  FROM  THE 
TABLES 

I  =  ~  (M  -  A).  (45) 

(1  +  15)0  +  30) 
/  300 

A 

(l  +  28  +  45)0-iQ+450 


I  (I  +  30) 

0  =  "loo  —  (10S) 


TABLES 


71 


TAPER 


INTERMEDIATE  VALUES 


A-J 


11 

I 

/ 

b 

o 

D 

logZ) 

d 

t 

z 

y 

Ic 

030 

30.00 

0021 

0021 

0  04* 

11459.18 

4.05915 

0.01 

15.00 

0.02 

30.00 

30.00 

35 

40.00 

0031 

003| 

007 

9822.17 

3.99221 

0.02 

20.00 

0.04 

40.00 

40.00 

40 

50.00 

0041 

005f 

010 

8594  41 

3.93422 

0.03 

25.00 

0.06 

50.00 

50.00 

45 

60.00 

005* 

008 

0  13* 

7639.49 

3.88306 

0.04 

30.00 

0.10 

60.00 

60.00 

50 

70.00 

007 

010* 

017* 

6875.56 

3.83731 

0.05 

35.00 

0.14 

70.00 

70.00 

55 

80.00 

008f 

0131 

022 

6250.53 

3.79592 

0.07 

40.00 

0.20 

80.00 

80.00 

100 

90.00 

010* 

016* 

027 

5729.70 

3.75813 

0.10 

45.00 

0.27 

90.00 

90.00 

05 

100.00 

012* 

020 

032* 

5288.99 

3.72337 

0.13 

50.00 

0.36 

100  00 

100.00 

10 

110.00 

014* 

024 

038* 

4911  25 

3.69119 

0.16 

55.00 

0.47 

110.00 

110.00 

15 

120.00 

016f 

0281 

045 

4583.89 

3.66123 

0.20 

60.00 

0.59 

120.00 

120.00 

20 

130.00 

0191 

032| 

052 

4297.45 

3.63321 

0.24 

65.00 

0.73 

130.00 

130.00 

25 

140.00 

022 

D37* 

059* 

4044.72 

3.60689 

0.29 

70.00 

0.89 

140.00 

140.00 

30 

150.00 

024f 

0  42f 

107* 

3820.09 

3.58207 

0.34 

75.00 

1.08 

149.99 

150.00 

35 

160.00 

027f 

0481 

1  16 

3619.11 

3.55860 

0.40 

80.00 

1.29 

159.99 

160.00 

40 

170.00 

030f 

0541 

125 

3438.25 

3.53634 

0.47 

85.00 

1.52 

169.99 

170.00 

45 

180.00 

034 

1  00* 

134* 

3274.63 

3.51516 

0.55 

90.00 

1.79 

179.98 

179.99 

50 

190.00 

037* 

107 

144* 

3125.89 

3.49497 

0.63 

94.99 

2.08 

189.98 

189.99 

55 

200.00 

0411 

1  13| 

1  55 

2990.10 

3.47569 

0.72 

99.99 

2.40 

199.98 

199.99 

200 

210.00 

045 

121 

206 

2865.64 

3.45722 

0.82 

104.99 

2.75 

209.97 

209.99 

05 

220.00 

049 

128* 

217* 

2751.16 

3.43952 

0.93 

109.99 

3.13 

219.96 

219.98 

10 

230.00 

053 

136* 

229* 

2645.51 

3.42251 

.05 

114.99 

3.55 

229.95 

229.98 

15 

240.00 

0571 

144| 

242 

2547.70 

3.40615 

.18 

119.99 

4.00 

239.94 

239.97 

20 

250.00 

lOlf 

1531 

255 

2456.89 

3.39039 

.32 

124.98 

4.49 

249.93 

249.97 

25 

260.00 

106* 

202 

308* 

2372.36 

3.37518 

.46 

129.98 

5.02 

259.92 

259.97 

30 

270.00 

nil 

2111 

322* 

2293.50 

3.36050 

.62 

134.98 

5.59 

269.90 

269.98 

35 

280.00 

1161 

220f 

337 

2219.74 

3.34630 

1.79 

139.97 

6.20 

279.88 

279.95 

40 

290.00 

1211 

2  301 

352 

2150.61 

3.33256 

1.97 

144.97 

6.86 

289.86 

289.94 

45 

300.00 

1  26* 

241 

407* 

.2085.69 

3.31925 

2.16 

149.96 

7.56 

299.84 

299.94 

50 

310.00 

1  32 

251* 

423* 

2024.62 

3.30634 

2.36 

154.96 

8.30 

309.81 

309.93 

55 

320.00 

1  37| 

3021 

440 

1967.06 

3.29382 

2.58 

159.95 

9.09 

319.78 

319.91 

300 

330.00 

143* 

313* 

457 

1912.72 

3.28165 

2.81 

164.94 

9.93 

329.74 

329.89 

05 

340.00 

149* 

325 

5  14* 

1861.35 

3.26983 

3.05 

169.93 

10.82 

339.70 

339.87 

10 

350.00 

155* 

337 

5  32* 

1812.70 

3.25833 

3.30 

174.93 

11.76 

349.66 

349.85 

15 

360.00 

201| 

3491 

551 

1766.58 

3.24713 

3.57 

179.92 

12.75 

359.61 

359.83 

20 

370.00 

2081 

401f 

610 

1722.78 

3.23623 

3.85 

184.90 

13.80 

369.55 

369.81 

25 

380.00 

2  15 

414* 

629* 

1681.16 

3.22561 

4.15 

189.89 

14.90 

379.49 

379.78 

30 

390.00 

221| 

427f 

649* 

1641.56 

3.21526 

4.47 

194.88 

16.07 

389.42 

389.75 

35 

400.00 

228f 

4411 

710 

1603.82 

3.20516 

4.80 

199.86 

17.29 

399.35 

399.72 

40 

410.00 

235f 

4551 

731 

1567.82 

3.19530 

5.14 

204.85 

18.57 

409.27 

409.69 

45 

420.00 

243 

509* 

752* 

1533.46 

3.18567 

5.50 

209.83 

19.91 

419.18 

419.65 

50 

430.00 

250* 

524 

814* 

1500.61 

3.17627 

5.87 

214.81 

21.31 

429.08 

429  61 

55 

440.00 

2581 

538| 

837 

1469.20 

3.16708 

6.26 

219.79 

22.78 

438.97 

439.56 

400 

450.00 

306 

554 

900 

1439.14 

3.15810 

6.67 

224.77 

24.31 

448.85 

449.51 

72 


THE   RAILROAD  TAPER 


u 


cS 


08 


US 


08 


02 


08 
dt, 


OS 


i  rrtT         FHH  " 

-    CO  "*<  O  <N  <N  O5  <M  CO  T-I  10  b-  to  T-I  Tt  CO 
O  I-H  <M  <M  (M  T-I  1-1  O  1O  CO  T-I  iO  CO  O  CO 
o    CO  CO  CO  CO  CO  CO  CO  CO  <M  CQ  <M  T-  i  T-H  i—  i  O 


co 

rt<  tO 


«H<«!'«MNrt!N,*m,-('-«ji               n|THn|-«n|-4iFH|M  n',-« 

T-  1  10  1^  CD  c^  »o  10  cq  o  co  rf  oi  i—  i  .co 

tO  iO  10  10  >0  "^  CO  <M  O  TH  <N  1C  CO  .CO 

T-ir-  i  O  O  '  O 


,_i  r*  T-I  o  O      'Ot-t 


Ot^-         O5  i—  I  t>- 
ioC^      iC^OCO 
0    <N  (N  (Tq  C^  (M  (M  1-1  I-H  1-1  1-1  O  O     'Ot-^i-H 


OO      'OOT-H  <M 


T-I  CO  (M  00  I-H  i—  i  t^  T—  i  (M 
CO  CO  CO  <M  C<J  I-H  10  ^  (M 


OOO      '  O  O  T-I  T-H  (M  CO 


i-i  i-HOOOOO      •  O  O  T-I  I-H  (M  C^  CO 


OOOOOOO      •  O  O  i—  i  i—  i  <N  <N  CO  CO 


.  n|-«m|-4i 

:^^fe^ 

OOOOOO      '  O  O  O  I-H  i—  i  (M  (M  CO  Tt< 


^  '~H     •  ?O  O  OCO  O5  GO  O:  -^  <N  co 

rH  rH        .  ^H  CO  iO  i-(  CO  O  CO  rH  ^  CO 

oOOOOO      •  O  O  O  I-H  T-H  (M  <N  CO  CO  rfn 


-^Oi      .  i—  i  CO  CO  Tt<  t^  Tf  rfi  CD  (M  rH  <M 
T-I  O      .T-I  <M-*O<N  »O^  »0  COr-H  10 
o    OOOO      •  O  O  O  T-I  T-I  r-t 


. 

»OOCO     .O5T-icOiOCOi-HGOGOiMOOI^O 

T-I    T-I    O  .   O    (M    CO    10   T-I    Tj<    O   CO    T-H    Th    (M    T=( 

o    OOO     'OOOOTHrH  <N  (N  COCOTJH  10 


OO     '  O  O  O  O  T-I  TH  T-H  (M  <N  CO  -^  ''t1  10 


. 
<N         .  ^  C^   CO  t^  TfH  Tfl  CO  (M  T-H   CO  OO  O  t^  rH 

O   .O  T-  i  (M  CO  O  T-I  CO  O  CO  O  CO  T-H  10  Tf 
O   '  O  O  O  O  O  T-I  T-H  <M  <M  CO  CO  TJ*  -^  10 


'  O  O  O  O  O  T-  i  T-H  rH  C^  C^l  CO  CO  -*i  >O  »O 


r 


TABLES 


73 


12 


8 


OiOiOOOOt- 


•^cocccocococoeococoeoeococcco 


TficOt^-iOOOC^OOiOO^OOit^-CO 
i-H^iO"tfC'OeOC<J<N<N<N'-H'-iT-t 


O  O  O  O  r-  1  1-1  <M  <M  CO  rJH  rti  iO  CO  !>•  OS 


i-H^fb-OCOCOOiC^iO 


o    oOr-ii-Hi—  ii—  i 


74 


THE   RAILROAD  TAPER 


dS 


08 

0?, 


08 

OS, 


68 


o 


08 


OS 


08 


08 


08 


OS 

o't, 


08 


-*f  ^  -^  Tf  •*  "^  CO  CO  <N  <M  T-H  O      '  O  <N 


. 

(MCO'^^COO-IOCOOOO'*   !LQ»OO 
CC  CO  CO  CO  CO  CO  CO  <N  (M  >—  i  O   '  O  i—  i  CO 


i--        .  -  -  -  - 


(N  CO  CO  CO  <N  (N  (M  T-H  ,-H  O   'Or-KMCO 


_ 

(M  O  CO  O  O5  CO  <M  •*  O      .  iO  CO  •*  OO  OO 
(M  CO  CO  <M  i—  i  O  "*  -—  i  •*      .  ^f  CO  CO  CO  "* 

<N  (M  (N  <N  (N  iM  I-H  1-1  O      'O^H<NCOTt< 


i—  i  <N  <M  i-H  i—  i  i—  I  i—  i  O      '  O  i—  I  <N  CO  rf  1C 


-    O  to  •"*!  l-^  to  O  T-H      .  COOO  t>- N  OQ.Ci  ' 
CO  CO  CO  <M  T-H  iO  CO      .  CO  T- i  O  O  O  O  < 

o     T-I  T-I  r_|  T-l  T-l  O  O       •  O  T-H  <M  CO  TjH  tO  CO 


-    00  <M  Oi  i—  i  t^  h-      .  I-H  Oi 

O  i—  i  O  O  Tti  (M      .  CO  O 

0       1—  I    T-t    T-H    T-l    O   O  '   O    1—  1 


-.- 

i  O  <M  i—  i 

r^  rti  »O 


. 

r-  1  oo  00  <N      .  t^  O  O  CO  (X)  >O  Oi  Oi 
itOrfiCOfN      .  (M  O  Tti  (N  i-H  ^-i  T-I  <M 

OOOOO      •O'-i 


. 

OO  .C<lr-i|^ 
CO  CO  <N  i—  i  .  (M  iO  (N 
OOOO  ;OOi-i 


^^H 
O  <M 


^T-^OCO      .GOfNCOOCOr-HCOt^rfcOiO 

CC,  ^  ^,    .  _  T?  r-l  10  CO  <N  i-H  T-l  (M  CO  >0 

°  ooo  ;  o  o  i—  i  -—  i  <N  co  •*  to  co  t^  oo 


oo    ;  o  o  i—  i  T-i  c<i  <N  co  TJ<  to  co  oo  oi 


o    O      ;  O  O  O  1-1  i-i  <N  CO  "*  iO  CO  l^  CX3  01 


oooo 


TABLES 


75 


oOOCiO^o^ooi>-coccc5^i>-i>- 

C^  ^D  ^^  Oi  O^  Oi  Oi  Oi  Oti  Oi  QO  00  t*^*  ^O 


'->Oo»0( 


CO  "^  CO  t^»  !>•  CO  O5  "^  T-H  ^^  t^»  t>»  ^^ 
l>»  *-O  "^  CO  0s-!  C^l  ^H  T—  •  j  CD  ^^ 


»OOi(M^»OiO 
O"—  iOi—  ilMT^i—i 


8888888^888888 


THE  RAILROAD  TAPER 


TAPER  1.     INTERMEDIATE  VALUES 


M 

I 

/ 

b 

o 

D 

logZ) 

d 

t 

z.  • 

V 

Ic 

0      1 

100 

30.00 

004$ 

004! 

009 

5729.62 

3.75813 

0.02 

15.00 

0.04 

30.00 

30.00 

05 

35.00 

005* 

006 

Oil! 

5288.89 

3.72336 

0.03 

17.50 

0.06 

35.00 

35.00 

10 

40.00 

006! 

007! 

0  14 

4911.12 

3.69118 

0.03 

20.00 

0.08 

40.00 

40.00 

15 

45.00 

007! 

0091 

0  16f 

4583.73 

3.66122 

0.04 

22.50 

0.10 

45.00 

45.00 

20 

50.00 

008| 

0111 

020 

4297.26 

3.63319 

0.05 

25.00 

0  13 

50.00 

50.00 

25 

55.00 

010 

0  13! 

023! 

4044.49 

3.60686 

0.06 

27.50 

0.16 

55.00 

55.0'J 

30 

60.00 

OllJ 

015! 

027 

3819.83 

3.58204 

0.08 

30.00 

0.20 

60.00 

60.00 

35 

65.00 

012f 

0  18 

030! 

3618.80 

3.55856 

0.09 

32.50 

0.24 

65.00 

65.00 

40 

70.00 

014J 

020! 

035 

3437.89 

3.53629 

0.11 

35.00 

0.29 

70.00 

70.00 

45 

75.00 

015f 

023f 

039! 

3274.21 

3.51511 

0.13 

37.50 

0.35 

75.00 

75.00 

50 

80.00 

om 

026! 

044 

3125.41 

3.49491 

0.15 

40.00 

0.41 

80.00 

80.00 

55 

85.00 

0191 

029! 

048! 

2989.55 

3.47561 

0.17 

42.50 

0.47 

85.00 

85.00 

200 

90.00 

021 

033 

054 

2865.02 

3.45713 

0.20 

45.00 

0.55 

90.00 

90.00 

05 

95.00 

023 

036! 

059! 

2750.45 

3.43940 

0.22 

47.50 

0.63 

95.00 

95.00 

10 

100.00 

025 

040 

105 

2644.71 

3.42238 

0.25 

50.00 

0.72 

100.00 

100.00 

15 

105.00 

027 

043! 

10! 

2546.80 

3.40599 

0.28 

52.50 

0.82 

105.00 

105.00 

20 

110.00 

0291 

047| 

17 

2455.89 

3.39021 

0.32 

55.00 

0.93 

109.99 

110.00 

25 

115.00 

031! 

052 

23! 

2371.25 

3.37498 

0.35 

57.50 

.05 

114.99 

115.00 

30 

120.00 

033! 

0561 

30 

2292.27 

3.36027 

0.39 

60.00 

.18 

119  99 

120.00 

35 

125.00 

0361 

100! 

36! 

2218.39 

3.34604 

0.44 

62  50 

.31 

124.99 

125.00 

40 

130.00 

038| 

1051 

44 

2149.12 

3.33226 

0.48 

65.00 

.46 

129.99 

130.00 

45 

135.00 

0411 

1101 

151! 

2084.06 

3.31891 

0.53 

67.49 

.62 

134.98 

134.99 

50 

140.00 

044 

1  15 

159 

2022.84 

3.30596 

0.58 

69.99 

.79 

139.98 

139.99 

55 

145.00 

046! 

120 

206! 

1965.11 

3  .  29339 

0.63 

72.49 

.97 

144.98 

144.99 

300 

150.00 

049| 

125! 

215 

1910.60 

3.28117 

0.69 

74.99 

2.16 

149.97 

149.99 

05 

155.00 

052^ 

131 

223! 

1859.05 

3.26929 

0.75 

77.49 

2.36 

154.97 

154.99 

10 

160.00 

055! 

136! 

232 

1810.21 

3.25773 

0.81 

79.99 

2.57 

159.97 

159.99 

15 

165.00 

058| 

1421 

240! 

1763.89 

3.24647 

0.88 

82.49 

2.80 

164.96 

164.98 

20 

170.00 

101! 

1481 

250 

1719.88 

3.23550 

0.95 

84.99 

3.05 

169.96 

169.98 

25 

175.00 

105 

153! 

259! 

1678.03 

3.22480 

1.02 

87.49 

3.30 

174.95 

174.98 

30 

180.00 

1081 

200| 

309 

1638.19 

3.21436 

1.10 

89.98 

3.57 

179.94 

179.98 

35 

185.00 

llf 

207 

318| 

1600.20 

3.20417 

1.18 

92.48 

3.86 

184.93 

184.97 

40 

190.00 

151 

2  13! 

329 

1563.95 

3.19422 

1.27 

94.98 

4.16 

189.92 

189.97 

45 

195.00 

18f 

220! 

339! 

1529.32 

3.18450 

1.36 

97.48 

4.47 

194.  U 

194.96 

50 

200.00 

22! 

227! 

350 

1496.19 

3.17499 

1.45 

99.97 

4.80 

199.90 

199.96 

55 

205.00 

261 

234! 

400f 

1464.49 

3.16569 

1.55 

102.47 

5.14 

204.89 

204.95 

TABLES  77 

TAPER   1.     INTERMEDIATE  VALUES  —  (Continued)     A  =  \ 


M 

I 

/ 

b 

0 

D 

logZ) 

d 

t 

z 

y 

Ic 

400 

210.00 

130 

242 

412 

1434.12 

3.15659 

1.65 

104.97 

5.50 

209.88 

209.95 

05 

215.00 

134 

249$ 

423$ 

1405.00 

3.14768 

1.76 

107.47 

5.87 

214.87 

214.95 

10 

220.00 

138 

257 

435 

1377.04 

3.13895 

1.87 

109.96 

6.26 

219.85 

219.94 

15 

225.00 

142 

3041 

4461 

1350.19 

3.13039 

1.98 

112.46 

6.67 

224.83 

224.93 

20 

230.00 

1461 

3  121 

459 

1324.39 

3.12201 

2.10 

114.95 

7.10 

229.82 

229.93 

25 

235.00 

150* 

321 

511$ 

1299.57 

3.11380 

2.23 

117.45 

7.54 

234.80 

234.92 

30 

240.00 

1  541 

3291 

524 

1275.68 

3.10574 

2.36 

119.94 

8.01 

239.77 

239.91 

35 

245.00 

1591 

337* 

5361 

1252.66 

3.09783 

2.49 

122.44 

8.49 

244.75 

244.90 

40 

250.00 

2031 

3461 

550 

1230.48 

3.09007 

2.63 

124.93 

8.99 

249.73 

249.89 

45 

255.00 

2081 

3551 

603$ 

1209.08 

3.08246 

2.77 

127.43 

9.51 

254.70 

254.88 

50 

260.00 

213 

404 

617 

1188.43 

3.07497 

2.92 

129.92 

10.05 

259.67 

259.87 

55 

265.00 

2171 

413 

6301 

1168.51 

3.06763 

3.08 

132.41 

10.60 

264.64 

264.85 

500 

270.00 

222$ 

422* 

645 

1149.25 

3.06041 

3.24 

134.91 

11.18 

269.61 

269.84 

05 

275.00 

227$ 

432 

659$ 

1130.60 

3.05331 

3.40 

137.40 

11.78 

274.57 

274.82 

10 

280.00 

232$ 

441$ 

714 

1112.61 

3.04634 

3.57 

139.89 

12.40 

279.53 

279.81 

15 

285.00 

237| 

4501 

7281 

1095.19 

3.03949 

3.75 

142.38 

13.04 

284.49 

284.80 

20 

290.00 

242f 

5011 

744 

1078.33 

3.03275 

3.94 

144.87 

13.71 

289.45 

289.78 

25 

295.00 

248 

5  1U 

759$ 

1062.00 

3.02612 

4.13 

147.36 

14.39 

294.40 

294.76 

30 

300.00 

2531 

5211 

815 

1046.16 

3.01960 

4.32 

149.85 

15.10 

299.35 

299.73 

35 

305.00 

258f 

532 

8301 

1030.81 

3.01318 

4.52 

152.34 

15.83 

304.30 

304.71 

40 

310.00 

3041 

5421 

847 

1015.93 

3.00686 

4.73 

154.83 

16.58 

309.24 

309.69 

45 

315.00 

309f 

"5531 

903$ 

1001.49 

3.00065 

4.94 

157.32 

17.36 

314.18 

314.66 

50 

320.00 

315| 

604$ 

920 

987.48 

2.99453 

5.16 

159.81 

18.16 

319.12 

319.63 

55 

325.00 

3211 

615$ 

9361 

973.86 

2.98850 

5.38 

162.30 

18.98 

324.06 

324.61 

600 

330.00 

327 

627 

954 

960.65 

2.98257 

5.61 

164.77 

19.83 

328.97 

329.57 

15 

345.00 

345 

7011 

10461 

923.18 

2.96529 

6.34 

172.22 

22.53 

343.73 

344.47 

-    30 

360.00 

4031 

7381 

1142 

888.73 

2.94877 

7.14 

179.67 

25.46 

358.44 

359.34 

45 

375.00 

4231 

8161 

1239$ 

856.95 

2.93296 

8.00 

187.10 

28.63 

373.10 

374.20 

700 

390.00 

443$ 

855| 

1339 

827.56 

2.91780 

8.92 

194.52 

32.05 

387.71 

389.03 

15 

405.00 

504$ 

9361 

14401 

800.33 

2.90327 

9.91 

201.93 

35.73 

402.25 

403.84 

30 

420.00 

5261 

10181 

1545 

775.05 

2.88933 

10.97 

209.33 

39.67 

416.73 

418  62 

78 


THE   RAILROAD  TAPER 


08 
do 


0*8 


dS 


03 


08 


02 

d°^ 


C5S 


o'S 


08 


OS 


08 

o't, 


OS 


H|N«|^                     Hh#H|Ni-t|NH|c>*rt|-*  mfa 

lO  Tt*  iO  CO  OO  I-H  »O  O  CO  CO  <M  i—  i  !  b- 

O  (M  CO  CO  (M  r-i  rfH  ,-1  (M  oo  CO  <M  .C^l 

CO  CO  CO  O  CO  CO  »O  iO  T^  CO  (N  ^H  •  ^H 


iO  iO  »O  kO  iO  iO  -^  CO  CO  (M  T-H        T-H 


I-H      ;  T-H  <N  rt< 


CO  CO  CO  CO  CO  CO  <N  I-H  r-H      ;^H(NCOto 


.  nWHwiHlNHNH'* 

<M(MCO»OOOC^OOT}H  .  O  i—  (  i—  i  Q  00 
»O  O  O  iO  CO  i—  i  CO  iO  .  O  T-I  CO  O  CO 
<N  CO  CO  <M  <M  (N  I-H  O  ;T-H<MCO»OCO 


<N  (M  <M  <M  I-H  T-H  O      ;  O  I-H  CO  ^  CO  1> 


OQTt<  <Ni-lO        .  l>  TjH  T-l 

I  -^        t  -^H   TJH  i/5 

O      -Oi-HCM^uOt^CX) 


.O'-HO 

i-i^Hr-HtOCO        .  Tt<  CO  CO  CO  1C  < 

T-IT-II-<OO    ;  o  I-H  <M  co  -^  co  t^  cs 


. 

O  »-H  -^  l>      .  CO  t^  O  C<J  CO  CO  <N  O  CO 
lOiO^C^       .COi—  li-Hi-HC^^i—  (LOCO 

oooo    ;  o  <-H  <N  co  Tt<  10  1>  oo  o 


ooo    ;  o  r-i  i—  i  c<<  co  »o  co  i>.  oi  I-H 


-  •'         .    -  --  M 


tX)>O(Mt^(NiOt^O5 
I-H  I-H  (M  CO  O  CO  T-I  O 

OO      'OOrH  (M  CO  -^  iOt^OOO(M 


«W  .  HIM  WHMW  H|MHN  H|M 

CO  .  CO  t^  O  i—  i  (N  i—  i  O  OO  Tf  O 

O  .I-H  CO  O  >OTjn  ^  iCOCO  r^ 

O  •  O  O  i-H  1-1  <N  CO  TH  CO  t^  O5  O  (N 


OOO  I-H  <M  CO  Tt<  10  COOOOii-HCO 


co  co  co  co  co  co  co  co  co 

COcOOiCNiOOOi—  i^t^ 

i—  IT—  IT—  IOQC>1CN 


TABLES 


79 


. 
*-H'^t|l>»OCO 


"e  co  ooo^H^*^' 

T 

[jj 

o 

O 

O      •* 

i  Oi  ^D  C^  C^l  CO  i-~i  CO  GO  i>-  CO  CD  CO 
^  ^  i^  T-H  TT  *O  OO  ^H  ^^  tO  *-O  Oi  t^»  !>• 

O  I^^C^T— irHOOiOt)OOQO001>*i>* 

tf 

q 


o 


CD 
CO 


THE   RAILROAD  TAPER 

TAPER  1|.     INTERMEDIATE  VALUES  A  =  | 


M 

I 

/ 

b 

0 

D 

log£> 

d 

t 

X 

y 

Ic 

130 

30.00 

006| 

006! 

013* 

3819.78 

3.58204 

0.03 

15.00 

0.06 

30.00 

30.00 

35 

33.33 

007! 

008 

015f 

3618.74 

3.55856 

0.03 

16.67 

0.07 

33.33 

33.33 

40 

36.67 

008| 

009* 

0181 

3437.82 

3.53628 

0.04 

18.33 

0.09 

36.67 

36.67 

45 

40.00 

009f 

0111 

021 

3274.13 

3.51509 

0.05 

20.00 

0.11 

40.00 

40.00 

50 

43.33 

OlOf 

013 

0231 

3125.32 

3.49489 

0.06 

21.67 

0.13 

43.33 

43.33 

55 

46.67 

Ollf 

015 

0261 

2989.45 

3.47559 

0.07 

23.33 

0.16 

46.67 

46.67 

200 

50.00 

013 

017 

030 

2864.90 

3.45726 

0.08 

25.00 

0.19 

50.00 

50.00 

05 

53.33 

0141 

019 

0331 

2750.32 

3.43938 

0.09 

26.67 

0.22 

53.33 

53.33 

10 

56.67 

015* 

0211 

0361 

2644.56 

3.42235 

0.10 

28.33 

0.26 

56.67 

56.67 

15 

60.00 

016! 

023! 

040* 

2546.64 

3.40597 

0.12 

30.00 

0.30 

60.00 

60.00 

20 

63.33 

0181 

026 

0441 

2455.70 

3.39018 

0.13 

31.67 

0.34 

63.33 

63.33 

25 

66.67 

019| 

028* 

0481 

2371.05 

3.37494 

0.15 

33.33 

0.38 

66.67 

66.67 

30 

70.00 

0211 

0311 

052* 

2292.04 

3.36022 

0.16 

35.00 

0.43 

70.00 

70.00 

33 

73.33 

022! 

034 

056| 

2218.13 

3.34599 

0.18 

36.67 

0.49 

73.33 

73.33 

40 

76.67 

024* 

036! 

1011 

2148.84 

3.33220 

0.20 

38.33 

0.55 

76.66 

76.67 

45 

80.00 

0261 

0  39! 

06 

2083.75 

3.31885 

0.22 

40.00 

0.61 

80.00 

80.00 

50 

83.33 

028 

0  42! 

1101 

2022.50 

3.30589 

0.24 

41.66 

0.68 

83.33 

83.33 

55 

86.67 

029f 

046 

115! 

1964.75 

3.29331 

0.27 

43.33 

0.75 

86.66 

86.66 

300 

90.00 

031* 

049* 

121 

1910.20 

3.28108 

0.29 

45.00 

0.82 

89.99 

89.99 

05 

93.33 

033* 

0521 

1261 

1858.62 

3.26919 

0.32 

46.66 

0.90 

93.33 

93.33 

10 

96.67 

035* 

0561 

131! 

1809.75 

3.25762 

0.35 

48.33 

0.99 

96.66 

96.66 

15 

100.00 

037* 

100 

137* 

1763.39 

3.24635 

0.38 

50.00 

.09 

99.99 

99.99 

20 

103.33 

039* 

1  031 

1431 

1719.34 

3.23536 

0.41 

51.66 

.19 

103.32 

103.33 

25 

106.67 

041* 

1071 

1491 

1677.45 

3.22465 

0.44 

53.33 

.29 

106.65 

106.66 

30 

110.00 

043f 

llf 

155* 

1637.56 

3.21420 

0.47 

54.99 

.40 

109.99 

109.99 

35 

113.33 

046 

151 

201! 

1599.53 

3.20399 

0.51 

56.66 

.52 

113.32 

113.33 

40 

116.67 

0481 

20 

2081 

1563.23 

3.19402 

0.55 

58.33 

1.64 

116.65 

116.66 

45 

120.00 

050* 

24* 

215 

1528.55 

3.18428 

0.59 

59.99 

1.77 

119.98 

119.99 

50 

123.33 

053 

281 

221! 

1495.37 

3.17475 

0.63 

61.66 

1.90 

123.31 

123.32 

55 

126.67 

055* 

1.331 

228! 

1463.61 

3.16543 

0.67 

63.33 

2.04 

126.64 

126.65 

400 

130.00 

058 

1  38 

236 

1433.18 

3.15630 

0.71 

64.99 

2.19 

129.97 

129.99 

05 

133.33 

100* 

1  421 

2431 

1404.00 

3.14737 

0.76 

66.66 

2.35 

133.30 

133.32 

10 

136.67 

1031 

1  47* 

250! 

1375.98 

3.13861 

0.81 

68.32 

2.51 

136.63 

136.65 

15 

140.00 

105| 

1  521 

258* 

1349.07 

3.13003 

0.86 

69.99 

2.68 

139.96 

139.99 

20 

143.33 

108* 

157f 

3061 

1323.20 

3.12163 

0.91 

71.65 

2.86 

143.29 

143.32 

25 

146.67 

111* 

2021 

3  141 

1298.31 

3.11338 

0.97 

73.32 

3.05 

146.62 

146.65 

TABLES  81 

TAPER   1J.     INTERMEDIATE  VALUES  —  (Continued)    A  =  f 


M 

I 

/ 

6 

0 

D 

logD 

d 

t 

X 

y 

Ic 

430 

150.00 

114} 

208} 

322* 

1274.35 

3.10529 

1.03 

74.98 

3.24 

149.94 

149.98 

35 

153.33 

1  17} 

212* 

330! 

1251.26 

3.09735 

.09 

76.65 

3.44 

153.27 

153.31 

40 

156.67 

120J 

219 

339} 

1229.00 

3.08955 

1.15 

78.31 

3.65 

156.60 

156.64 

45 

160.00 

123| 

224! 

348 

1207.52 

3.08189 

.21 

79.98 

3.87 

159.92 

159.97 

50 

163.33 

126} 

230| 

356! 

1186.79 

3.07437 

.28 

81.64 

4.10 

163.25 

163.30 

55 

166.67 

129i 

236* 

405f 

1166.78 

3.06699 

.35 

83.31 

4.33 

166.58 

166.63 

500 

170.00 

132* 

242* 

415 

1147.43 

3.05973 

.42 

84.97 

4.57 

169.90 

169.96 

05 

173.33 

135| 

248* 

424} 

1128.69 

3.05257 

.49 

86.64 

4.83 

173.22 

173.29 

10 

176.67 

139 

254| 

433! 

1110.61 

3.04556 

.57 

88.30 

5.09 

176.55 

176.62 

15 

180.00 

142} 

301} 

443* 

1093.09 

3.03865 

.65 

89.96 

5.36 

179.87 

179.95 

20 

183.33 

145| 

307| 

453} 

1076.12 

3.03186 

.73 

91.63 

5.64 

183.19 

183.28 

25 

186.67 

149} 

314 

503} 

1059.68 

3.02517 

1.81 

93.29 

5.93 

186.51 

186.60 

30 

190.00 

152| 

319! 

513* 

1043.74 

3.01859 

1.90 

94.95 

6.23 

189.83 

189.93 

35 

193.33 

156} 

3  27J 

523! 

1028.28 

3.01211 

1.99 

96.62 

6.54 

193.15 

193.25 

40 

196.67 

200 

334} 

534} 

1013.28 

3.00573 

2.08 

98.28 

6.86 

196.47 

196.59 

45 

200.00 

203| 

341} 

545 

998.72 

2.99944 

2.17 

99.94 

7.19 

199.79 

199.71 

50 

203.33 

207* 

348} 

555! 

984.59 

2.99326 

2.27 

101.61 

7.53 

203.10 

203.24 

55 

206.67 

2  11} 

355* 

606! 

970.85 

2.98715 

2.37 

103.27 

7.88 

206.42 

206.57 

600 

210.00 

215 

403 

618 

957.51 

2.98114 

2.47 

104.93 

8.24 

209.73 

209.89 

15 

220.00 

226! 

425f 

652* 

919.64 

2.96362 

2.80 

109.92 

9.39 

219.66 

219.86 

30 

230.00 

239} 

449} 

728* 

884.74 

2.94682 

3.15 

114.90 

10.64 

229.58 

229.83 

45 

240.00 

252i 

513! 

806 

852.48 

2.93068 

3.53 

119.88 

12.00 

239.49 

239.79 

700 

250.00 

305* 

539* 

845 

822.58 

2.91518 

3.94 

124.85 

13.47 

249.38 

249.75 

15 

260.00 

319* 

606 

925* 

794.80 

2.90026 

4.38 

129.82 

15.05 

259.26 

259.70 

30 

270.00 

333f 

633! 

1007* 

768.93 

2.88589 

4.85 

134.79 

16.76 

269.12 

269.64 

45 

280.00 

348| 

702* 

1051 

744.80 

2.87204 

5.36 

139.76 

18.58 

278.95 

279.57 

800 

290.00 

404 

732 

1136 

722.24 

2.85868 

5.90 

144.72 

20.53 

288.76 

289.49 

15 

300.00 

420 

802* 

12  22f 

701.11 

2.84579 

6.47 

149.67 

22.61 

298.54 

299.40 

30 

310.00 

436} 

834} 

1310* 

681.30 

2.83334 

7.08 

154.62 

24.83 

308.29 

309.29 

45 

320.00 

453 

907 

1400 

662.70 

2.82132 

7.73 

159.56 

27.18 

318.02 

319.17 

900 

330.00 

510* 

940* 

1451 

645.20 

2.80969 

8.42 

164.49 

29.68 

327.70 

329.04 

15 

340.00 

528* 

1015 

15  43* 

628.72 

2.79846 

9.14 

169.42 

32.32 

337.34 

338.88 

30 

350.00 

546| 

1050! 

1637* 

613.19 

2.78760 

9.90 

174.34 

35.11 

346.94 

348.72 

45 

360.00 

605* 

1127* 

1733 

598.53 

2.77709 

10.70 

179.25 

38.06 

356.50 

358.54 

1000 

370.00 

625 

1205 

1830 

584.69 

2.76693 

11.55 

184.15 

41.16 

366.01 

368.33 

15 

380.00 

645| 

1242! 

1928* 

571.61 

2.75710 

12.44 

189.04 

44.43 

375.47 

378.09 

30 

390.00 

705} 

13  23} 

2028* 

559.24 

2.74760 

13.37 

193.92 

47.85 

384.87 

387.82 

82 


THE   RAILROAD  TAPER 


8 


O 


Wg 


08 


uS 


u'S 


US 


US 


0      CO  l^  !>.  t>-  t>  O  «O  O  rJH  CO  r-  1         •  i—  ( 


o  -^  *o  to 

00  0000  00  00  OOt^  CO  to -^  CO 


rt|N  rt|M  H|N  rH|M  HM  H 

CD  CO  CO  OS  C^  Tt<  CO  i—  1  CD  O      '  O5  i-i 

-^  OrHO»o  w^  to^co    ;coco 


iO  CO  O  O  10  "3  rt<  CO  <N  '-i      •  T-H  CO 


Tfl  IO  ^O  Tfl  Ttl  Tj<  CO  <N  i—  1         •  l—t  CO  »O 


H|M        He*        IH|N        H«  ' 

O5  CO  Ttn  rt<  T-H  t^  O  <M      •  i—  i 

^  oo  toco  toi^  i-3    -<N 


8S8S8801  iSSsSSS 

CO  CO  CO  <N  <M  i— i  T-H      •  i— i  <M  Tfi  CO  00 


r 

-    CO  rJH  O5  CO  -<f  "^       '  CO  Oi  OO  OO  i—  (  to 
TH  <N  I-H  O  CO  to      ;  O  -H  TJH  <M  <M  C^ 


O5  CO  CO  CO  to      •  -^  i— i  T-H  <N  CO  T-H  O 
CO  "*  CO  T-H  "tfi      ;  10  O  (M  iO  CO  CO  CO 

THT-H^HT-HO     •  O  <M  CO  ^  CO  00  O 


I 

t^CiOOcO 
OOtOCO 


!—  i  i-H  O  O      •OT-( 


H|M  H|M  H|M  H|M  --' 

OOO      •  O  I-H  <M  CO  to'co  OO  O  (M 


OO      •  O  i—  (  C<l  CO  ••*  tO  l^  O5  i—  i  CO 


O      •  O  O  I-H  C^l  CO  "*  CD  OO  O  (M  Tt^ 


o       -OOrH  T-I  (N'*  to  COOOO<N  »O 


TABLES 


83 


18 


8 


O5  OS  OS 
i—  i^t^ 


OiO(NO05O 

i-H  rH  (M  CO  CO  1O 


o 


O  T-H 


co  t^  co 

O  CO  O  O  r-J 


'—  lOfNTf 

t^~  T-4  CO  OS  ^^  "^ 

lOOOiOiOOOi—  ( 


g 


qnSS^^^j 
i— i  OS  00  t^  CO  i 


OOrHCO^COOOOCOCOOS 


OCOCOOSCO 

I—  1   T—  I   T—  1  T—  1    C^ 


i— i  I-H  i— i  <M  C^  C^  ( 


888888888888 

<MCOTfitocOl>.OOOSO'— icNCO 


84 


THE   RAILROAD  TAPER 

TAPER  2.     INTERMEDIATE  VALUES  A  =  l 


Af 

I 

/ 

b 

0 

D 

log/) 

d 

t 

X 

y 

k 

200 

30.00 

009 

009 

018 

2864.86 

3.45710 

0.04 

15.00 

0.08 

30.00 

30.00 

05 

32.50 

010 

010* 

020* 

2750.28 

3.43938 

0.05 

16.25 

0.09 

32.50 

32.50 

10 

35.00 

OlOf 

012 

022! 

2644.51 

3.42234 

0.05 

17.50 

0.11 

35.00 

35.00 

15 

37.50 

Oil! 

013* 

025* 

2546.58 

3.40596 

0.06 

18.75 

0.13 

37.50 

37.50 

20 

40.00 

0121 

015* 

028 

2455.64 

3.39016 

0.07 

20.00 

0.15 

40.00 

40.00 

25 

42.50 

014 

016| 

030! 

2370.98 

3.37493 

0.08 

21.25 

0.17 

42.50 

42.50 

30 

45.00 

015 

018! 

033! 

2291.97 

3.36021 

0.09 

22.50 

0.20 

45.00 

45.00 

35 

47.50 

016* 

020* 

036! 

2218.05 

3.34597 

0.10 

23.75 

0.22 

47.50 

47.50 

40 

50.00 

017* 

022! 

i040 

2148.75 

3.33219 

0.11 

25.00 

0.25 

50.00 

50.00 

45 

52.50 

018* 

024| 

043* 

2083.65 

3.31882 

0.12 

26.25 

0.28 

52.50 

52.50 

50 

55.00 

019f 

027 

046! 

2022.39 

3.30586 

0.13 

27.50 

0.32 

55.00 

55.00 

55 

57.50 

021* 

029 

050* 

1964.52 

3.29326 

0.14 

28.75 

0.35 

57.50 

57.50 

300 

60.00 

022* 

031* 

054 

1910.07 

3.28105 

0.16 

30.00 

0.39 

60.00 

60.00 

05 

62.50 

024 

033! 

057f 

1858.47 

3.26916 

0.17 

31.25 

0.43 

62.50 

62.50 

10 

65.00 

025* 

036* 

lOlf 

1809.59 

3.25758 

0.19 

32.50 

0.48 

65.00 

65.00 

15 

67.50 

026f 

039 

105! 

1763.21 

3.24630 

0.20 

33.75 

0.53 

67.50 

67.50 

20 

70.00 

028J 

041! 

110 

1719.15 

3.23531 

0.22 

35.00 

0.58 

70.00 

70.00 

25 

72.50 

030 

044* 

114* 

1677.25 

3.22460 

0.24 

36.25 

0.63 

72.50 

72.50 

30 

75.00 

031* 

047* 

118! 

1637.35 

3.21414 

0.26 

37.50 

0.69 

75.00 

75.00 

35 

77.50 

033* 

050 

123* 

1599.30 

3.20393 

0.28 

38.75 

0.75 

77.50 

77.50 

40 

80.00 

034| 

053* 

128 

1562.98 

3.19395 

0.30 

40.00 

0.81 

79.99 

80.00 

45 

82.50 

036* 

056* 

132! 

1528.28 

3.18420 

0.32 

41.25 

0.88 

82.49 

82.50 

50 

85.00 

038* 

059* 

137! 

1495.08 

3.17437 

0.34 

42.50 

0.95 

84.99 

85.00 

55 

87.50 

040* 

102* 

142! 

1463.31 

3.16534 

0.37 

43.75 

.02 

87.49 

87.50 

400 

90.00 

042 

104 

148 

1432  86 

3.15620 

0.39 

45.00 

.10 

89.99 

90.00 

05 

92.50 

044 

109* 

153* 

1403.66 

3.14726 

0.42 

46.24 

.18 

92.49 

92.50 

10 

95.00 

045f 

1  13 

158! 

1375.62 

3.13850 

0.45 

47.49 

.27 

95.99 

95.00 

15 

97.50 

047f 

116* 

204* 

1348.69 

3.12991 

0.48 

48.74 

.36 

97.49 

97.50 

20 

100.00 

049f 

20* 

210 

1322.80 

3.12149 

0.51 

49.99 

1.45 

99.98 

99.99 

25 

102.50 

052 

23! 

215! 

1297.87 

3.11323 

0.54 

51.24 

1.55 

102.48 

102.49 

30 

105.00 

054 

27! 

221! 

1273.89 

3.10513 

0.57 

52.49 

1.65 

104.98 

104.99 

35 

107.50 

056* 

31* 

227! 

1250.77 

3.09718 

0.60 

53.74 

1.76 

107.48 

107.49 

40 

110.00 

058* 

135! 

234 

1228.48 

3.08937 

0.63 

54.99 

1.87 

109.98 

109.99 

45 

112.50 

100* 

139! 

240* 

1206.98 

3.08170 

0.67 

56.24 

1.98 

112.47 

112.49 

50 

115.00 

102f 

144 

246! 

1186.22 

3.07416 

0.71 

57.49 

2.10 

114.97 

114.99 

55 

117.50 

105* 

148 

253* 

1166.18 

3.06677 

0.75 

58.74 

2.23 

117.47 

117.49 

500 

120.00 

107* 

152* 

300 

1146.80 

3.05949 

0.79 

59.99 

2.36 

119.96 

119.99 

05 

122.50 

1  10 

156! 

306! 

1128.03 

3.05232 

0.83 

61.24 

2.49 

122.46 

122.49 

10 

125.00 

112* 

201* 

3  13! 

1109.91 

3.04529 

0.87 

62.48 

2.63 

124.96 

124.99 

TABLES  85 

TAPER  2.     INTERMEDIATE   VALUES  —  (Continued)        A  =  l 


u 

I 

/ 

b 

0 

D 

logD 

d 

t 

X 

V 

Ic 

5  15 

127.50 

14f 

206 

3201 

1092.35 

3.03836 

0.91 

63.73 

2.77 

127.45 

127.49 

20 

130.00 

m 

2101 

328 

1075.35 

3.03155 

0.96 

64.98 

2.92 

129.95 

129.98 

25 

132.50 

20 

2151 

3351 

1058.87 

3.02484 

.00 

66.23 

3.08 

132.44 

132.48 

30 

135.00 

22* 

2201 

3421 

1042.89 

3.01823 

.05 

67.48 

3.24 

134.94 

134.98 

35 

137.50 

251 

225 

3501 

1027.39 

3.01174 

.10 

68.73 

3.41 

137.44 

137.48 

40 

140.00 

27f 

2301 

358 

1012.35 

3.00490 

.15 

69.98 

3.58 

139.93 

139.98 

45 

142.50 

30* 

2351 

4051 

997.76 

2.99903 

.21 

71.22 

3.75 

142.42 

142.48 

50 

145.00 

331 

240* 

4  131 

983.58 

2.99281 

.26 

72.47 

3.93 

144.92 

144.97 

55 

147.50 

361 

245* 

4211 

969.80 

2.98668 

.32 

73.72 

4.12 

147.41 

147.47 

600 

150.00 

39 

251 

430 

'  956.41 

2.98064 

.37 

74.97 

4.32 

149.90 

149.97 

15 

157.50 

147f 

307* 

4551 

918.39 

2.96303 

.55 

78.71 

4.94 

157.37 

157.47 

30 

165.00 

57 

324f 

5211 

883.34 

2.94613 

.75 

82.45 

5.61 

164.84 

164.96 

45 

172.50 

206* 

342| 

5491 

850.92 

2.92989 

.97 

86.19 

6.34 

172.32 

172.44 

700 

180.00 

2  16* 

401* 

618 

820.84 

2.91426 

2.20 

89.93 

7.14 

179.77 

179.92 

15 

187.50 

226| 

421 

6471 

792.86 

2.89920 

2.44 

93.67 

8.00 

187.22 

187.39 

30 

195.00 

237| 

4411 

7  181 

766.79 

2.88468 

2.71 

97.41 

8.93 

194.66 

194.86 

45 

202.50 

248* 

5021 

7501 

742.43 

2.87066 

2.99 

101  .  14 

9.92 

202.10 

202.34 

800 

210.00 

300 

524 

824 

719.63 

2.85711 

3.29 

104.87 

10.98 

209.52 

209.81 

15 

217.50 

3  m 

546* 

8581 

698.26 

2.84402 

3.62 

108.60 

12.12 

216.93 

217.26 

30 

225.00 

324 

609f 

9331 

678.18 

2.83134 

3.96 

112.33 

13.33 

224.33 

224.72 

45 

232.50 

336| 

633f 

10101 

659.29 

2.81908 

4.32 

116.05 

14.62 

231.72 

232.18 

900 

240.00 

349* 

658* 

1048 

641.49 

2.80719 

4.71 

119.78 

15.99 

239.10 

239.63 

15 

247.50 

402f 

724 

11261 

624.69 

2.79566 

5.11 

123.49 

17.44 

246.46 

247.07 

30 

255.00 

4  16* 

7501 

12061 

608.83 

2.78450 

5.54 

127.21 

18.97 

253.80 

254.51 

45 

262.50 

430* 

8  17} 

12471 

593.82 

2.77365 

5.99 

130.92 

20.59 

261.12 

261.92 

1000 

270.00 

445 

845 

1330 

579.61 

2.76314 

6.47 

134.63 

22.30 

268.42 

269.36 

15 

277.50 

459| 

9  13* 

14131 

566.14 

2.75292 

6.97 

138.33 

24.11 

275.71 

276.76 

30 

285.00 

5  15 

942f 

14  571 

553.36 

2.74301 

7.49 

142.03 

26.00 

282.96 

284.16 

45 

292.50 

530* 

10  121 

15  431 

541.22 

2.73337 

8.04 

145.72 

27.99 

290.20 

291.54 

1100 

300.00 

546* 

1043* 

1630 

529.69 

2.72402 

8.62 

149.41 

30.08 

297.40 

298.92 

15 

307.50 

602| 

1115 

17171 

518.73 

2.71494 

9.23 

153.09 

32.26 

304.58 

306.28 

30 

315.00 

619* 

1147J 

18061 

508.29 

2.70611 

9.86 

156.77 

34.55 

311.72 

313.64 

45 

322.50 

636* 

12201 

18561 

498.36 

2.69754 

10.52 

160.44 

36.94 

318.84 

319.50 

1200 

330.00 

654 

1254 

1948 

488.88 

2.68920 

11.20 

164.10 

39.44 

325.91 

328.29 

15 

337.50 

7111 

1328* 

20401 

479.86 

2.68111 

11.92 

167.76 

42.04 

332.95 

335.59 

30 

345.00 

730 

14  031 

21  331 

471.25 

2.67325 

12.66 

171.41 

44.76 

339.95 

342.88 

45 

352.50 

748* 

1439| 

22281 

463.05 

2.66563 

13  44 

175.05 

47.59 

346.90 

350.15 

1300 

360.00 

807* 

1516* 

2324 

455.22 

2.65822 

14.25 

178.68 

.  50.51 

353.81 

357.40 

86 


THE  RAILROAD  TAPER 


o 


0 
02 


do 


OS 


08 


8 


08 


ds 

Win 


t>-  1O  00  b-  0  00  rH  OS  <N  O  CO      • 

CO  O  rH  1-1  O  <N  ^  CO  (M  iC  O      • 

CO  O5  OS  O5  O5  00  t»  CO  iO  CO  CQ      ' 


*    CD  t^  <N  CO  OO  O5  >O  U5  T-H      '  (M  <N 

10  I-H  c^j  i-t  Tt<  o  I-H  o  Ttn    ;ioo 
o   xo  CO  CO  CO  iC  >O  •"*!  CO  i—  i     •  T-I  TP 


i-*      •  i—  i  CO  1C 


CO  CO  CO  CO  CO  <M  rH      -T-H 


H«        I  W|^H|N  HN 

10  O  »o  co  »o  o    ;  i—  1  10  Tt^  10 

o    <N  CO  <N  (N  »-<  *-i      -r- 


<N  (N  C<l  rH  O        •  rH  <N  Tf  CD  00  O 


»    ^OCO>O 

(M  <N  i-H  ^ 

o    rH  i—  i  TH  O 


OOC5O 

CO  C^  rH  CO  O 

O  <M  CO  iO  l>  O:  (N 


OOO      •  O  rH  CO  •'f  CD  OO  O  CO 


GOC4 
(M  <M 

OO 


OOOt^lM 
CO  10  (N  i-t  <M 

O  rH  <M  CO  1C  1>  O5 


O        -O  rH  rHCO  ^  CO  GO  O  <N  »O 


•  O  O  rH  oq  CO  iO  O  00  O  CO  O 


TABLES 


87 


K 


J? 


O1  0s  IO  O5 


^  O^  C^  "^  QO  T-H  C*Q  O^  '^ 
QOiOCOOOi—  iCOi—  ICOCO 
T—  *OO5OOO01>-l>-OCO 


COCOO(NO<NCOOOI>.i—  (CO 
OirH»OOOOOI^<NCOCOt^<N 

»—  !OOOl>.tOI>l>»OcDl^O 
O5  C^  ^*  i^  CO  *O  t^-  T-H  CO  C^l  OS 


o    O  --I  <N  00  iO  I>  O  CO  JOO  TfH 


CO  CO  O5  (N  iO  00  i—  i 


THE   RAILROAD  TAPER 
TAPER  2J.        INTERMEDIATE  VALUES 


M 

I 

/ 

b 

o 

D 

logD 

d 

t 

X 

y 

Ic 

230 

30.00 

oil} 

0  111 

022* 

2291.93 

3.36020 

0.05 

15.00 

0.10 

30.00 

30.00 

35 

32.00 

012 

012! 

024f 

2218.01 

3.34596 

0.06 

16.00 

0.11 

32.00 

32.00 

40 

34.00 

013 

0141 

0271 

2148.70 

3.33218 

0.06 

17.00 

0.13 

34.00 

34.00 

45 

36.00 

014 

0  15J 

029! 

2083.60 

3.31881 

0.07 

18.00 

0.15 

36.00 

36.00 

50 

38.00 

015 

0171 

0321 

2022.34 

3.30585 

0.08 

19.00 

0.17 

38.00 

38.00 

55 

40.00 

016 

019 

035 

1964.57 

3.29327 

0.09 

20.00 

0.19 

40.00 

40.00 

300 

42.00 

017 

020! 

0371 

1910.00 

3.28103 

0.09 

21.00 

0.21 

42.00 

42.00 

05 

44.00 

018} 

022* 

0401 

1858.40 

3.26914 

0.10 

22.00 

0.23 

44.00 

44.00 

10 

46.00 

019} 

024* 

043f 

1809.51 

3.25756 

0.11 

23.00 

0.26 

46.00 

46.00 

15 

48.00 

020| 

0261 

0461 

1763.13 

3.24628 

0.12 

24.00 

0.29 

48.01) 

48.00 

20 

50.00 

021| 

0281 

050 

1719.06 

3.23529 

0.13 

25.00 

0.32 

50.00 

50.00 

25 

52.00 

023 

0301 

0531 

1677.16 

3.22457 

0.15 

26.00 

0.35 

52.00 

52.00 

30 

54.00 

024} 

032* 

0561 

1637.25 

3.21411 

0.16 

27.00 

0.38 

54.00 

54.00 

35 

56.00 

025* 

034f 

1001 

1599.19 

3.20390 

0.17 

28.00 

0.41 

56.00 

56.00 

40 

58.00 

026f 

037 

1031 

1562.86 

3.19392 

0.18 

29.00 

0.45 

58.00 

58.00 

45 

60.00 

028 

039* 

107* 

1528.16 

3.18417 

0.20 

30.00 

0.49 

60.00 

60.00 

50 

62.00 

029* 

041| 

HI 

1494.95 

3.17463 

0.21 

31.00 

0.53 

62.00 

62.00 

55 

64.00 

031 

0441 

151 

1463.17 

3.16529 

0.23 

32.00 

0.57 

64.00 

64.00 

400 

66.00 

032* 

046! 

191 

1432.71 

3.15616 

0.24 

33.00 

0.62 

66.00 

66.00 

05 

68.00 

034 

0491 

231 

1403.50 

3.14721 

0.26 

34.00 

0.67 

68.00 

68.00 

10 

70.00 

035* 

052 

27* 

1375.45 

3.13844 

0.28 

35.00 

0.72 

69.99 

70.00 

15 

72.00 

037 

054f 

31! 

1348.50 

3.12985 

0.29 

36.00 

0.77 

71.99 

72.00 

20 

74.00 

038* 

057! 

361 

1322.60 

3.12143 

0.31 

37.00 

0.83 

73.99 

74.00 

25 

76.00 

040} 

100* 

40! 

1297.67 

3.11316 

0.33 

38.00 

0.89 

75.99 

76.00 

30 

78.00 

041! 

103* 

451 

1273.67 

3.10506 

0.35 

39.00 

0.95 

77.99 

78.00 

35 

80.00 

043* 

106* 

150 

1250.54 

3.09710 

0.37 

40.00 

.01 

79.99 

80.00 

40 

82.00 

045J 

109* 

154! 

1228.25 

3.08929 

0.40 

40.99 

.08 

81.99 

82.00 

45 

84.00 

047 

12! 

159! 

1206.73 

3.08161 

0.42 

41.99 

.15 

83.99 

84.00 

,50 

86.00 

048f 

16 

204! 

1185.95 

3.07407 

0.44 

42.99 

.22 

85.99 

86.00 

55 

88.00 

0  50* 

191 

209! 

1165.90 

3.06666 

0.47 

43.99 

.30 

87.99 

88.00 

500 

90.00 

052* 

22* 

2  15 

1146.50 

3.05937 

0.49 

44.99 

.37 

89.98 

90.00 

05 

92.00 

054* 

25! 

220} 

1127.72 

3.05220 

0.52 

45.99 

.45 

91.98 

91.99 

10 

94.00 

0561 

29* 

225! 

1109.58 

3.04516 

0.54 

46.99 

.54 

93.98 

93.99 

15 

96.00 

0581 

133 

231} 

1092.01 

3.03823 

0.57 

47.99 

1.63 

95.98 

95.99 

.      20 

98.00 

1001 

136* 

236! 

1074.99 

3.03140 

0.60 

48.99 

1.72 

97.98 

97.99 

25 

100.00 

1021 

1401 

2  42* 

1058.50 

3.02469 

0.63 

49.99 

1.81 

99.98 

99.99 

30 

102.00 

1041 

144 

248} 

1042.50 

3.01808 

0.66 

50.99 

1.91 

101.97 

101.99 

35 

104.00 

106* 

147f 

254} 

1026.98 

3.01156 

0.69 

51.99 

2.01 

103.97 

103.99 

40 

106.00 

108* 

151f 

300} 

1011.92 

3.00515 

0.72 

52.99 

2.11 

105.97 

105.99 

TABLES  89 

TAPER  2i     INTERMEDIATE  VALVES— (Continued)   A  =  U 


M     \      I 

/ 

b 

0 

D 

log-Z) 

d 

t 

X 

V 

fc 

545 

108.00 

1  10| 

1  55* 

306* 

997.31 

2.99883 

0.76 

53.99 

2.22 

107.97 

107.99 

'50 

110.00 

1  13 

159* 

3  12* 

983.11 

2.99260 

0.79 

54.98 

2.33 

109.96 

109.99 

55 

112.00 

1  15i 

203* 

3  18! 

969.31 

2.98646 

0.83 

55.98 

2.45 

111.96 

111.99 

600 

114.00 

ii7* 

207! 

325* 

955.90 

2.98041 

0.86 

56.98 

2.57 

113.95 

113.98 

15 

120.00 

124* 

220* 

345 

917.82 

2.96276 

0.98 

59.98 

2.95 

119.94 

119.98 

30 

126.00 

1  31| 

234 

405! 

882.70 

2.94581 

.11 

62.97 

3.36 

125.93 

125.97 

45 

132.00 

139* 

248 

427* 

850.19 

2.92952 

.24 

65.97 

3.81 

131.91 

131.97 

700 

138.00 

147 

302! 

449!' 

820.03 

2.91383 

.39 

68.96 

4.30 

137.89 

137.96 

15 

144.00 

155i 

318 

513* 

791.97 

2.89871 

.55 

71.96 

4.83 

143.87 

143.95 

30 

150.00 

203| 

333! 

537* 

765.80 

2.88412 

.72 

74.95 

5.40 

149.84 

149.94 

45 

156.00 

212| 

350* 

602! 

741.34 

2.87002 

.90 

77.94 

6.01 

155.81 

155.93 

800 

162.00 

221* 

407* 

628! 

718.43 

2.85638 

2.09 

80.93 

6.67 

161.78 

161.91 

15 

168.00 

231 

424! 

655! 

696.93 

2.84319 

2.29 

8,3.92 

7.37 

167.73 

167.89 

30 

174.00 

240| 

443 

723! 

676.73 

2.83042 

2.51 

86.91 

8.12 

173.69 

173.88 

45 

180.00 

250! 

501! 

752* 

657.71 

2.81803 

2.74 

89.90 

8.92 

179.63 

179.86 

900 

186.00 

301 

521* 

822* 

639.77 

2.80602 

2.99 

92.88 

9.77 

185.57 

185.84 

15 

192.00 

3  11* 

541* 

852! 

622.83 

2.79437 

3.25 

95.86 

10.68 

191.51 

191.82 

30 

198.00 

322* 

602 

924* 

606.81 

2.78305 

3.52 

98.84 

11.64 

197.43 

197.79 

45 

204.00 

333* 

623* 

956! 

591.64 

2.77206 

3.81 

101.82 

12.65 

203.34 

203.75 

1000 

210.00 

345 

645 

1030 

577.26 

2.76137 

4.12 

104.80 

13.71 

209.25 

209.70 

15 

216.00 

356f 

707* 

1104* 

563.61 

2.75098 

4.44 

107.78 

14.84 

215.14 

215.65 

30 

222.00 

408f 

730* 

1139* 

550.64 

2.74087 

4.77 

110.75 

16.03 

221.02 

221.60 

45 

228.00 

42H 

754 

1215* 

538.30 

2.73102 

5.12 

113.72 

17.27 

226.89 

227.55 

1100 

234.00 

434 

818* 

1252* 

526.56 

2.72145 

5.49 

116.69 

18.58 

232.74 

233.49 

15    240.00 

447 

843 

1330 

515.38 

2.71213 

5.88 

119.65 

19.96 

238.59 

239.42 

30 

246.00 

500* 

908* 

1408! 

504.71 

2.70304 

6.28 

122.61 

21.40 

244.42 

245.36 

45 

252.00 

513! 

934* 

1448* 

494.54 

2.69420 

6.70 

125.57 

22.90 

250.23 

251.29 

1200 

258.00 

527| 

1001* 

1528! 

484.82 

2.68558 

7.14 

128.52 

24.47 

256.02 

257.20 

15 

264.00 

541! 

1028* 

16  10* 

475.54 

2.67719 

7.60 

131.47 

26.11 

261.79 

263.09 

30 

270.00 

556* 

1056* 

1652* 

466.67 

2.66901 

8.08 

134.42 

27.82 

267.54 

268.98 

45 

276.00 

611 

1124! 

1735! 

458.19 

2.66105 

8.58 

137.36 

29.61 

273.27 

274.87 

1300 

282.00 

626 

1153! 

18  19! 

450.06 

2.65327 

9.09 

• 

140.30 

31.47 

278.98 

280.75 

15 

288.00 

641* 

1223* 

1904! 

442.29 

2.64571 

9.63 

143.23 

33.40 

284.66 

286.61 

30 

294.00 

657* 

1253* 

1950! 

434.85 

2.63834 

10.19 

146.16 

35.40 

290.32 

292.47 

45 

300.00 

713* 

1324^ 

2037* 

427.71 

2.63115 

10.76 

149.08 

37.49 

295.95 

298.32 

1400 

306.00 

729* 

1355! 

2125* 

420.87 

2.62415 

11.36 

152.00 

39.65 

301.55 

304.15 

15 

312.00 

746 

14  27| 

2213! 

414.31 

2.61733 

11.98 

154.91" 

41.89 

307.12 

309.96 

30 

318.00 

802! 

1500* 

2303* 

408.04 

2.61070 

12.63 

157.81 

44.21 

312.66 

315.76 

45 

324.00 

820 

1523! 

2353! 

402.01 

2.60424 

13.29 

160.71 

46.61 

318.16 

321.55 

1500 

330.00 

837* 

1607* 

2445 

396.23 

2.59795 

13.98 

163.60 

49.09 

323.63 

327.33 

90 


THE   RAILROAD  TAPER 


d8 


d8 


qg 

OS. 


08 


08 


OiOOCOOOCOiOCOOltO 
CO  O  <N  i— i  •*  O  O  "^  O  i— i 

GOO5O5O500OOt^»O'^(N 


•  U5  £H  CN  S  ^        O  C^ 
iOCOCO»OXOTt<COi-i 


. g^ %% %%$ 

0    ^TjHT^^COlMi-l 


<N  C4 

cocceoco(Mi-i    :  I-H  co  "3  oo 


<NCO(N>OO 


T— (  CO  t^  ^      •  t^  10  i— i -^  CO  CO 

rH  T-t  t— I  O       !  T— I  <N  -^  CO  00  T-H 


O(MCO»O1>O(M 


COC5      -^  ^OCOCO  ^  TH  O 


OO       .OrH 


O  T-  (  <N  CO  »O  t^  O5  (N  iO 


i-i  (M-^  COOOOCO  CO 


i-H  ^H  ^-i  (M  W  <M 


TABLES 


91 


OOOiOiOil>-iO'—  iiOO 


^D  CD  OS  OS  OS  OS  GO  t^» 
COCOGO'—  i  -^  l>  O  CO 


OOOSOSOSGOt^iOT-HCO 


0 


CO 


I—"*  Oi  ^D  ^H  O5  i"H  •^ 
QC  ^^  00  00  ^D  CO  OO  CO  Oi  *^O 


00  *O  *O  GO  OS  C^l  T™H  r-H 

t^-iOCOCO^fGOCOOS 


iH  I-H  i—  i  (M  (M  <M 


SwScoScoScooco 


1—  1   1—  1   T-H    1—  1   T-H 


92 


THE   RAILROAD  TAPER 
TAPER  2|.     INTERMEDIATE  VALUES 


M 

I 

/ 

•6 

0 

D 

logZ) 

d 

t 

X 

y 

k 

300 

30.00 

013£ 

0  13^ 

027 

1909.97 

3.28103 

0.06 

15.00 

0.12 

30.00 

30.00 

05 

31.67 

0  14^ 

014! 

0291 

1858.37 

3.26913 

0.07 

15.83 

0.13 

31.67 

31.67 

10 

33.33 

0  151 

016J 

031| 

1809.47 

3.25755 

0.07 

16.67 

0.15 

33.33 

33.33 

15 

35.00 

0161 

0  18i 

034£ 

1763.09 

3.24627 

0.08 

17.50 

0.17 

35.00 

35.00 

20 

36.67 

0171 

019£ 

036| 

1719.02 

3.23528 

0.09 

18.33 

0.18 

36.67 

36.67 

25 

38.33 

0181 

021 

0391 

1677.10 

3.22456 

0.09 

19.17 

0.20 

38.33 

38.33 

30 

40.00 

0191 

022| 

042 

1637.19 

3.21410 

0.10 

20.00 

0.22 

40.00 

40.00 

35 

41.67 

0201 

024^ 

044! 

1599.13 

3.20388 

0.11 

20.83 

0.24 

41.67 

41.67 

40 

43.33 

021£ 

0261 

047! 

1562.80 

3.19390 

0.12 

21.67 

0.27 

43.33 

43.33 

45 

45.00 

022§ 

0281 

050! 

1528.09 

3.18415 

0.13 

22.50 

0.29 

45.00 

45.00 

50 

46.67 

023f 

030 

053! 

1494.88 

3.17461 

0.14 

23.33 

0.32 

46.67 

46.67 

55 

48.33 

024f 

032 

056! 

1463.09 

3.16527 

0.15 

24.17 

0.35 

48.33 

48.33 

400 

50.00 

026 

034 

1  00 

1432.63 

3.15613 

0.16 

25.00 

0.38 

50.00 

50.00 

05 

51.67 

0271 

036 

1031 

1403.41 

3.14715 

0.17 

25.83 

0.41 

51.66 

51.67 

10 

53.33 

028| 

0381 

106! 

1375.35 

3.13841 

0.18 

26.66 

0.44 

53.33 

53.33 

15 

55.00 

029f 

040^ 

101 

1348.41 

3.12982 

0.20 

27.50 

0.48 

55.00 

55.00 

20 

56.67 

031 

042! 

113! 

1322.50 

3.12140 

0.21 

28.33 

0.51 

56.66 

56.67 

25 

58.33 

032£ 

044! 

171 

1297.56 

3.11313 

0.22 

29.16 

0.55 

58.33 

58.33 

35 

60.00 

033f 

047J 

21 

1273.56 

3.10502 

0.24 

30.00 

0.59 

60.00 

60.00 

30 

61.67 

0351 

049£ 

24! 

1250.42 

3.09706 

0.25 

30.83 

0.63 

61.66 

61.67 

40 

63.33 

036£ 

0521 

28! 

1228.12 

3.08924 

0.27 

31.66 

0.67 

63.33 

63.33 

45 

65.00 

038 

054f 

132! 

1206.59 

3.08156 

0.28 

32.50 

0.72 

64.99 

65.00 

50 

66.67 

039£ 

0571 

136! 

1185.81 

3.07402 

0.30 

33.33 

0.77 

66.66 

66.67 

55 

68.33 

041 

059! 

140! 

1165.74 

3.06660 

0.31 

34.16 

0.82 

68.33 

68.33 

500 

70.00 

042£ 

102£ 

145 

1146.34 

3.05931 

0.33 

35.00 

0.87 

69.99 

70.00 

05 

71.67 

044 

105J 

1491 

1127.55 

3.05214 

0.35 

35.83 

0.92 

71.66 

71.66 

10 

73.33 

045| 

108 

153! 

1109.41 

3.04509 

0.37 

36.66 

0.97 

73.32 

73.33 

15 

75.00 

0471 

1  11 

1581 

1091.83 

3.03816 

0.39 

37.49 

1.03 

74.99 

75.00 

20 

76.67 

049 

13 

202! 

1074.80 

3.03133 

0.41 

38.33 

1.09 

76.66 

76.66 

25 

78.33 

050J 

16! 

2071 

1058.30 

3.02461 

0.43 

39.16 

1.15 

78.32 

78.33 

30 

80.00 

0521 

19! 

212 

1042.29 

3.01799 

0.45 

39.99 

1.22 

79.99 

80.00 

35 

81.67 

054 

22f 

216! 

1026.76 

3.01147 

0.47 

40.83 

1.28 

81.65 

81.66 

40 

83.33 

055f 

26 

221! 

1011.69 

3.00505 

0.49 

41.66 

1.35 

83.32 

83.33 

45 

85.00 

057£ 

291 

226f 

997.07 

2.99873 

0.52 

42.49 

1.42 

84.98 

84.99 

50 

87.67 

0591 

132^ 

231! 

982.86 

2.99249 

0.54 

43.32 

1.49 

86.65 

86.66 

55 

88.33 

loll 

135J 

236! 

969.04 

2.98634 

0.56 

44.16 

1.57 

88.31 

88.33 

TABLES  93 

TAPER  2|.     INTERMEDIATE  VALUES  —  (Continued)  A  =  l% 


M 

/ 

/ 

b 

0 

D 

logD 

i 

t 

X 

V 

Ic 

600 

90.00 

103 

139 

242 

955.63 

2.98029 

0.59 

44.99 

1.65 

89.98 

89.99 

15 

95.00 

1  08| 

1  49i 

258! 

917.51 

2.96261 

0.67 

47.49 

1.90 

94.97 

94.99 

30 

100.00 

114f 

200! 

315 

882.35 

2.94564 

0.76 

49.98 

2.18 

99.96 

99.99 

45 

105.00 

1  21 

2  11! 

332! 

849.80 

2.92932 

0.85 

52.48 

2.48 

104.96 

104.99 

700 

110.00 

1274 

2  234 

351 

819.59 

2.91360 

0.95 

54.98 

2.80 

109.95 

109.98 

15 

115.00 

134! 

236 

410! 

791.48 

2.89844 

.06 

57.47 

3.15 

114.93 

114.98 

30 

120.00 

141! 

248| 

430 

765.26 

2.88381 

.18 

59.97 

3.53 

119.92 

119.97 

45 

125.00 

148^ 

302* 

4  501 

740.75 

2.86967 

.31 

62.46 

3.94 

124.90 

124.96 

800 

130.00 

156 

3  16 

512 

717.78 

2.85599 

.44 

64.96 

4.38 

129.88 

129.95 

15 

135.00 

203! 

330§ 

534! 

696.22 

2.84275 

.58 

67.45 

4.85 

134.86 

134.95 

30 

140.00 

2  llf 

345J 

557 

675.95 

2.82991 

.73 

69.94 

5.36 

139.84 

139.94 

45 

145.00 

220 

400| 

620! 

656.86 

2.81747 

.89 

72.44 

5.90 

144.81 

144.93 

900 

150.00 

228| 

416* 

645 

638.84 

2.80539 

2.06 

74.93 

6.47 

149.77 

149.91 

15 

155.00 

237! 

433 

7  10! 

621.82 

2.79366 

2.24 

77.42 

7.08 

154.74 

154.90 

30 

160.00 

246! 

449| 

736 

605.72 

2.78227 

2.43 

79.91 

7.73 

159.70 

159.88 

45 

165.00 

255| 

507! 

802! 

590.46 

2.77119 

2.63 

82.39 

8.42 

164.65 

164.86 

1000 

170.00 

305 

525 

830 

575.98 

2.76041 

2.84 

84.88 

9.14 

169.60 

169.84 

15 

175.00 

314! 

543§ 

858! 

562.23 

2.74991 

3.06 

87.37 

9.90 

174.54 

174.82 

30 

180.00 

3  24! 

6  02! 

927 

549.16 

2.73970 

3.29 

89.85 

10.70 

179.47 

179.79 

45 

185.00 

335 

6  21! 

956! 

536.71 

2.72974 

3.53 

92.33 

11.55 

184.40 

184.76 

1100 

190.00 

345£ 

641± 

1027 

524.86 

2.72004 

3.79 

94.81 

12.44 

189.32 

189.73 

15 

195.00 

356* 

702 

1058! 

513.56 

2.71059 

4.06 

97.29 

13.37 

194.24 

194.69 

30 

200.00 

407! 

722! 

1130 

502.77 

2.70137 

4.34 

99.77 

14.35 

199.14 

199.65 

45 

205.00 

418£ 

744i 

1202! 

492.47 

2.69238 

4.63 

102.24 

15.37 

204.03 

204.61 

1200 

210.00 

430 

806 

1236 

482.62 

2.68361 

4.94 

104.72 

16.44 

208.92 

209.56 

15 

215.00 

441| 

828^ 

13  10! 

473.20 

2.67504 

5.26 

107.19 

17.56 

213.79 

214.51 

30 

220.00 

453f 

85U 

1345 

464.18 

2.66669 

5.59 

109.65 

18.73 

218.65 

219.46 

45 

225.00 

506 

9  14! 

1420! 

455.54 

2.65853 

5.93 

112.12 

19.95 

223.50 

224.40 

1300 

230.00 

518J 

938§ 

1457 

447.26 

2.65056 

6.29 

114.58 

21.22 

228.34 

229.33 

15 

235.00 

531* 

1003 

1534! 

439.32 

2.64278 

6.66 

117.04 

22.54 

233.17 

234.25 

30 

240.00 

5441 

10  271 

16  12 

431.71 

2.63519 

7.05 

119.50 

23.91 

237.98 

239.17 

45 

245.00 

557! 

10  53J 

16  50-1 

424.40 

2.62778 

7.45 

121.95 

25.34 

242.77 

244.08 

1400 

250.00 

611 

11  19 

1730 

417.38 

2.62053 

7.87 

124.40 

26.82 

247.55 

248.99 

15 

255.00 

624f 

1145§ 

1810! 

410.63 

2.61345 

8.30 

126.85 

28.36 

252.30 

253.89 

30 

260.00 

638! 

12  12! 

1851 

404.15 

2.60654 

8.74 

129.29 

29.95 

257.04 

258.78 

45 

265.00 

653 

12  39f 

1932! 

397.92 

2.59980 

9.20 

131.7$ 

31.60 

261.77 

263  67 

1500 

270.00 

707* 

1307J 

2015 

391.93 

2.59321 

9.68 

134.17 

33.31 

266.47 

268.54 

30 

280.00 

737! 

14041 

2142 

380.62 

2.58049 

10.69 

139.03 

36.90 

275.81 

278.26 

1600 

290.00 

808 

1504 

23  12 

370.15 

2.56838 

11.76 

143.87 

40.74 

285.05 

287.94 

30 

300.00 

839| 

16  05! 

2445 

360.45 

2.55684 

12.90 

148.68 

44.82 

294.19 

297.58 

94 


THE  RAILROAD  TAPER 


TAPER  2|.    CURVATURE  INCREASING  2°  00' 

EVERY  30  FT.  A =2 


Deflections  to 

Transit 

at 

B.C. 

c.c. 

C.C. 

C.C. 

C.C. 

C.C. 

C.C. 

C.C. 

C.C. 

C.C. 

2°  00' 

4°  00' 

6°  00' 

8°  00' 

10°  00' 

12°  00' 

14°  00' 

16°  00' 

18°  00' 

20°  00' 

I 

0 

30 

60 

90 

120 

150 

180 

210 

240 

270 

0 

0  18 

045 

124 

215 

3  18 

433 

600 

739 

930 

30 

018 

036 

121 

218 

327 

448 

621 

806 

1003 

60 

1  03 

036 

054 

157 

312 

439 

618 

809 

10  12 

90 

2  12 

139 

054 

1  12 

233 

406 

551 

748 

957 

120 

345 

306 

215 

1  12 

1  30 

309 

500 

703 

9  18 

150 

542 

457 

400 

251 

130 

1  48 

345 

554 

8  15 

180 

803 

7  12 

609 

454 

327 

148 

206 

421 

648 

210 

1048 

951 

842 

721 

548 

403 

206 



224 

457 

240 

1357 

1254 

1139 

1012 

833 

642 

439 

224 

242 

270 

1730 

1621 

1500 

1327 

1142 

945 

736 

5  15 

242 

M 

I 

o 

D 

log£> 

d 

t 

2 

y 

Ic 

0        / 

400 

30 

036 

1432.55 

3.15611 

0.08 

15.00 

0.16 

30.00 

30.00 

600 

60 

148 

955.35 

2.98016 

0.31 

29.99 

0  79 

59.99 

60.00 

800 

90 

336 

717.12 

2.85559 

0.78 

44.98 

2.20 

89.96 

89.98 

1000 

120 

600 

574.71 

2.75945 

1.57 

59.95 

4.70 

119.85 

119.95 

1200 

150 

900 

480.42 

2.68162 

2.74 

74.87 

8.63 

149.60 

149.85 

1400 

180 

1236 

413.90 

2.61690 

4.39 

89.73 

14.25 

179.07 

179.64 

1600 

210 

1648 

364.96 

2.56225 

6.57 

104.50 

21.87 

208.08 

209.24 

1800 

240 

21  36 

328.02 

2.51590 

9.38 

119.12 

31.77 

236.44 

238.54 

2000 

270 

2700 

299.71 

2.47670 

12.87 

133.54 

44.13 

263.76 

267.43 

TABLES  95 

TAPER  2|.    INTERMEDIATE  VALUES          A =2 


M 

I 

/ 

b 

o 

D 

logD 

d 

t 

X 

V 

Ic 

0      1 

400 

30.00 

018 

018 

036 

1432.55 

3.15611 

0.08 

15.00 

0.16 

30.00 

30.00 

05 

31.25 

019 

019J 

0381 

1403.33 

3.14716 

0.09- 

15.62 

0.17 

31.25 

31.25 

10 

32.50 

019| 

021 

040f 

1375.26 

3.13838 

0.09 

16.25 

0.19 

32.50 

32.50 

'      15 

33.75 

020J 

022J 

043 

1348.31 

3.12979 

0.10 

16.87 

0.20 

33.75 

33.75 

20 

35.00 

021f 

023! 

045} 

1322.40 

3.12136 

0.11 

17.50 

0.22 

35.00 

35.00 

25 

36.25 

022! 

0251 

048 

1297.45 

3.11309 

0.11 

18.12 

0.24 

36.25 

36.25 

30 

37.50 

023f 

027 

050! 

1273.44 

3.10498 

0.12 

18.75 

0.26 

37.50 

37.50 

35 

38.75 

024! 

028} 

0531 

1250.30 

3.09701 

0.13 

19.37 

0.28 

38.75 

38.75 

40 

40.00 

025f 

0301 

056 

1227.99 

3.08919 

0.14 

20.00 

0.30 

40.00 

40.00 

45 

41.25 

026f 

032 

058! 

1206.46 

3.08151 

0.15 

20.62 

0.32 

41.25 

41.25 

50 

42.50 

027f 

033f 

OH 

1185.66 

3.07396 

0.15 

21.25 

0.34 

42.50 

42.50 

55 

43.75 

029 

035| 

104} 

1165.59 

3.06655 

0.16 

21.87 

0.37 

43.75 

43.75 

500 

45.00 

030 

037^- 

107} 

1146.18 

3.05925 

0.17 

22.50 

0.39 

45.00 

45.00 

05 

46.25 

03H 

0391 

10* 

1127.38 

3.05207 

0.18 

23.12 

0.42 

46.25 

46.25 

10 

47.50 

0321 

0411 

Wi 

1109.23 

3.04502 

0.19 

23.75 

0.44 

47.50 

47.50 

15 

48.75 

033} 

0431 

16! 

1091.64 

3.03808 

0.20 

24.37 

0.47 

48.75 

48.75 

20 

50.00 

034! 

0451 

20 

1074.60 

3.03125 

0.21 

25.00 

0.50 

50.00 

50.00 

25 

51.25 

036 

0471 

231 

1058.09 

3.02452 

0.22 

25.62 

0.54 

51.25 

51.25 

30 

52.50 

037 

049} 

1  26} 

1042.08 

3.01790 

0.24 

26.25 

0.57 

52.50 

52.50 

35 

53.75 

038} 

051| 

130 

1026.54 

3.01138 

0.25 

26.87 

0.60 

53.75 

53.75 

40 

55.00 

039! 

053! 

1  33| 

1011.46 

3.00495 

0.26 

27.50 

0.63 

55.00 

55.00 

45 

56.25 

041 

056 

137 

996.83 

2.99862 

0.28 

28.12 

0.67 

56.25 

56.25 

50 

57.50 

0421 

0581 

140} 

982.61 

2.99238 

0.29 

28.75 

0.71 

57.50 

57.50 

55 

58.75 

043! 

100} 

1441 

968.78 

2.98623 

0.30 

29.37 

0.75 

58.75 

58.75 

600 

60.00 

045 

103 

148 

955.35 

2.98016 

0.31 

29.99 

0.79 

59.99 

60.00 

15 

63.75 

0491 

1101 

159} 

917.20 

2.96246 

0.36 

31.87 

0.91 

63.74 

63.75 

30 

67.50 

053! 

1  18 

211! 

882.00 

2.94547 

0.41 

33.74 

1.05 

67.49 

67.50 

45 

71.25 

0581 

126 

2241 

849.41 

2.92912 

0.46 

35.62 

1.21 

71.24 

71.25 

700 

75.00 

103 

134} 

237} 

819.15 

2.91336 

0.51 

37.49 

1.38 

74.98 

75.00 

15 

78.75 

108 

1431 

2511 

790.99 

2.89817 

0.57 

39.36 

1.56 

78.73 

78.74 

30 

82.50 

1  13 

152} 

305} 

764.72 

2.88350 

0.64 

41.24 

1.76 

82.47 

82.49 

45 

86.25 

1  18} 

202 

320} 

740.15 

2.86932 

0.71 

43.11 

1.97 

86.22 

86.24 

800 

90.00 

124 

212 

336 

717.12 

2.85559 

0.78 

44.98 

2.20 

89.96 

89.98 

15 

93.75 

129! 

2221 

352 

695.50 

2.84230 

0.86 

46.85 

2.45 

93.70 

93.73 

30 

97.50 

135f 

233 

408f 

675.17 

2.82941 

0.95 

48.73 

2.71 

97.44 

97.48 

45 

101.25 

141! 

244 

425! 

656.01 

2.81691 

1.04 

50.60 

2.99 

101.18 

101.23 

96 


THE  RAILROAD  TAPER 


TAPER  2|.     INTERMEDIATE  VALUES  —  (Continued)     A  =  2 


M 

I 

/ 

b 

o 

D 

log/) 

d 

t 

X 

V 

Ic 

0      / 

900 

105.00 

148 

255} 

•4  43} 

637.91 

2.80476 

1.13 

52.47 

3.30 

104.92 

104.97 

15 

108.75 

154J 

307} 

'501| 

620.81 

2.79296 

1.23 

54.34 

3.62 

108.66 

108.71 

30 

112.50 

2.01 

3  19$ 

520} 

604.63 

2.78149 

.34 

56.21 

3.96 

112.39 

112.46 

45 

116.25 

208 

332 

540 

589.28 

2.77032 

.45 

58.08 

4.32 

116.12 

116.21 

1000 

120.00 

2  15 

345 

600 

574.71 

2.75945 

.57 

59.95 

4.70 

119.85 

119.9* 

15 

123.75 

2221 

3581 

620} 

560.86 

2.74885 

.69 

61.82 

5.11 

123.58 

123.69 

30 

127.50 

229$ 

4  12 

641} 

547.69 

2.73853 

.82 

63  68 

5.54 

127.31 

127.43 

45 

131.25 

2371 

426 

7031 

535.14 

2.72847 

.96 

65.55 

6.00 

131.03 

131.17 

1100 

135.00 

245 

440} 

725} 

523.17 

2.71864 

2.10 

67.42 

6.48 

134.75 

134.91 

15 

138.75 

253 

4551 

7481 

511.75 

2.70906 

2.25 

69.28 

6.98 

138.47 

138.65 

30 

142  50 

301 

510} 

811} 

500.84 

2.69970 

2.41 

71.15 

7.51 

142.18 

142.39 

45 

146.25 

309} 

526 

835} 

490.41 

2.69056 

2.57 

73.01 

8.06 

145.89 

146.12 

1200 

150.00 

3  18 

542 

900 

480.42 

2.68162 

2.74 

74.87 

8.63 

149.60 

149.85 

15 

153.75 

326| 

558} 

9251 

470.86 

2.67289 

2.92 

76.74 

9.23 

153.30 

153.58 

30 

157.50 

335} 

615 

950} 

461.69 

2.66435 

3.10 

78.60 

9.86 

157.00 

157.31 

45 

161.25 

344f 

632 

1016| 

452.91 

2.65601 

3.30 

80.46 

10.52 

160.69 

161.04 

1300 

165.00 

354 

649} 

1043} 

444.47 

2.64784 

3.50 

82.32 

11.20 

164.37 

164.76 

15 

168.75 

403} 

7071 

11  10| 

436.37 

2.63985 

3.71 

84.17 

11.92 

168.06 

168.48 

30 

172.50 

413 

725} 

1138} 

428.59 

2.63204 

3.93 

86.03 

12.67 

171.74 

172.20 

45 

176.25 

423 

744 

1207 

421.10 

2.62439 

4.15 

87.89 

13.45 

175.41 

175.92 

1400 

180.00 

433 

803 

1236 

413.90 

2.61690 

4.39 

89.73 

14.25 

179.07 

179.64 

15 

183.75 

4431 

8221 

1305} 

406.96 

2.60955 

4.63 

91.59 

15.09 

182.73 

183.35 

30 

187.50 

453} 

842 

1335} 

400.29 

2.60237 

4.88 

93.44 

15.96 

186.38 

187.06 

45 

191.25 

5041 

902 

14061 

393.86 

2.59534 

5.14 

95.29 

16.86 

190.02 

190  76 

1500 

195.00 

515 

932} 

1437} 

387.66 

2.58845 

5.41 

97.14 

17.79 

193.65 

194.46 

30 

202.50 

5371 

1004} 

1541! 

375.90 

2.57507 

5.97 

100.82 

19.76 

200.89 

201.86 

1600 

210.00 

600 

1048 

1648 

364.96 

2.56225 

6.57 

104.50 

21.87 

208.08 

209.24 

30 

217.50 

623} 

1133 

1756} 

354.76 

2.54993 

7.21 

108.18 

24.12 

215.25 

216.60 

1700 

225.00 

6  48      12  19} 

1907} 

345.23 

2.53811 

7.89 

111.85 

26.52 

222.37 

223.94 

30 

232.50 

7  13      13  07} 

2020} 

336.33 

2.52677 

8.61 

115.49 

29.07 

229.43 

231.26 

1800 

240.00 

7  39    1  13  57 

2136 

328.02 

2.51590 

9.38 

119.12 

31.77 

236.44 

238.54 

30 

247.50 

805} 

1448 

2253} 

320.22 

2.50545 

10.18 

122.76 

34.63 

243.38 

245.83 

1900 

255.00 

833 

1540} 

24  13} 

312.94 

2.49546 

11.03 

126.38 

37.64 

250.26 

253.07 

30 

262.50 

901 

1634} 

25  35} 

306  10 

2.48586 

11.92 

129.98 

40.81  257.06 

260.27 

2000 

270.00 

930 

1730 

2700 

299.71 

2.47670 

12.87 

133.54 

44.13 

263.76 

267.43 

TABLES 


97 


TAPER  3.     CURVATURE   INCREASING  2°  30' 

EVERY  30  FT.  A 


Deflections  to 

rr 

1  rjinsit 

at 

B.C. 

c.c. 

C.C. 

C.C. 

C.C. 

C.C. 

C.C. 

C.C. 

C.C. 

2°  30' 

5°  00' 

7°  30' 

10°  00' 

12°  30' 

15°  00' 

17°  30' 

20°  00 

'   22°  30' 

I 

0 

30 

60 

90 

120 

150 

180 

210 

240 

0 

0        , 

022* 

056i 

1  45 

248f 

407* 

54H 

730 

933| 

30 

022* 

045 

141i 

252* 

4  18f 

600 

756* 

1007* 

60 

1  18f 

045 

1  07* 

226i 

400 

548f 

752* 

10  in 

90 

245 

203f 

1  07* 

130 

3  Hi 

507* 

718f 

945 

120 

44l| 

352* 

248| 

1  30 

152* 

356i 

615 

848| 

150 

707* 

6  11J 

500 

333| 

152* 

215 

4411 

722* 

180 

1003f 

900 

741i 

607* 

4  18| 

215 

237* 

526i 

210 

1330 

1218f 

10  52* 

91U 

715 

503f 

237* 

300 

240 

1726i 

1607* 

1433f 

1245 

10  4H 

822* 

5481 

300 

M 

I 

o 

D 

logZ) 

d 

t 

X 

V 

U 

0        1 

500 

30 

045 

1146.11 

3.05923 

0.10 

15.00 

0.20 

30.00 

30.00 

730 

60 

215 

764.47 

2.88336 

0.39 

29.99 

0.98 

59.99 

60.00 

1000 

90 

430. 

574.12 

2.75900 

0.98 

44.97 

2.75 

89.94 

89.98 

1230 

120 

730 

460.55 

2.66328 

1.96 

59.91 

5.88 

119.77 

119.91 

1500 

150 

11  15 

385.68 

2.58623 

3.43 

74.80 

10.77 

149.37 

149.76 

1730 

180 

1545 

333.21 

2.52270 

5.48 

89.59 

17.78 

178.54 

179.41 

2000 

210 

2100 

295.04 

2.46988 

8.20 

104.22 

27.25 

207.01 

208.78 

2230 

240       2700 

266.74 

2.42609 

11.69 

118.63 

39.49 

234.42 

237.72 

98 


THE   RAILROAD  TAPER 
TAPER  3.    INTERMEDIATE  VALUES 


M 

I 

/ 

6 

o 

D 

log£> 

d 

t 

X 

y 

Ic 

500 

30.00 

022* 

022* 

045 

1146.11 

3.05923 

0.10 

15.00 

0.20 

30.00 

30.00 

05 

'31.00 

023* 

023! 

047} 

1127.30 

3.05204 

0.10 

15.50 

0.21 

31.00 

31.00 

10 

32.00 

024} 

025} 

049* 

1109.15 

3.04499 

0.11 

16.00 

0.22 

32.00 

32.00 

15 

33.00 

025} 

026f 

052 

1091.56 

3.03805 

0.12 

16.50 

0.24 

33.00 

33.00 

20 

34.00 

026} 

028} 

054* 

1074.51 

3.03121 

0.12 

17.00 

0.26 

34.00 

34.00 

25 

35.00 

027 

030 

057 

1058.00 

3.02449 

0.13 

17.50 

0.28 

35.00 

35.00 

30 

36.00 

028 

031* 

059* 

1041.98 

3.01786 

0.14 

18.00 

0.29 

36.00 

36.00 

35 

37.00 

029 

033 

1  02 

1026.44 

3.01133 

0.15 

18.50 

0.31 

37.00 

37.00 

40 

38.00 

030 

034* 

104* 

1011.35 

3.00490 

0.15 

19.00 

0.33 

38.00 

38.00 

45 

39.00 

031 

036} 

107} 

996.71 

2.99857 

0.16 

19.50 

0.35 

39.00 

39.00 

50 

40.00 

032 

038 

1  10 

982.49 

2.99233 

0.17 

20.00 

0.37 

40.00 

40.00 

55 

41.00 

033i 

039* 

1  12! 

968.66 

2.98617 

0.18 

20.50 

0.40 

41.00 

41.00 

600 

42.00 

034} 

041} 

1  15* 

955.23 

2.98011 

0.19 

21.00 

0.42 

42.00 

42.00 

15 

45.00 

037* 

046f 

124} 

917.05 

2.96239 

0.21 

22.50 

0.49 

45.00 

45.00 

30 

48.00 

041 

052* 

133* 

881.83 

2.94538 

0.24 

24.00 

0.57 

48.00 

48.00 

45 

51.00 

044* 

058| 

143} 

849.23 

2.92903 

0.28 

25.50 

0.66 

51.00 

51.00 

700 

54.00 

048} 

1  05} 

153* 

818.95 

2.91326 

0.31 

26.99 

0.76 

53.99 

54.00 

15 

57.00 

052} 

nil 

204 

790.77 

2.89805 

0.35 

28.49 

0.87 

56.99 

57.00 

30 

60.00 

056} 

1  18f 

2  15 

764.47 

2.88336 

0.39 

29.99 

0.98 

59.99 

60.00 

45 

63.00 

100* 

1  26 

226* 

739.88 

2.86916 

0.44 

31.49 

1.11 

62.99 

63.00 

800 

66.00 

1  04f 

133! 

238* 

716.82 

2.85541 

0.48 

32.99 

1.24 

65.98 

66.00 

15 

69.00 

109} 

141* 

250| 

695.17 

2.84209 

0.53 

34.49 

1.39 

68.98 

69.00 

30 

72.00 

1  14 

149* 

303* 

674.81 

2.82918 

0.59 

35.99 

1.55 

71.98 

71.99 

45 

75.00 

1  18! 

158 

316! 

655.61 

2.81665 

0.64 

37.48 

1.72 

74.97 

74.99 

900 

78.00 

123f 

206f 

330* 

637.48 

2.80447 

0.70 

38.98 

1.90 

77.97 

77.99 

15     81.00 

128f 

216 

344! 

620.35 

2.79264 

0.77 

40.48 

2.09 

80.96 

80.99 

30     84.00 

1  34      2  25* 

359* 

604.13 

2.78113 

0.84 

41.98 

2.30 

83.95 

83.99 

45     87.00 

1  39*  i  2  35 

414* 

588.74 

2.76992 

0.91 

43.47 

2.52 

86.95 

86.98 

1000     90.00 

1  45      2  45 

430 

574.12 

2.75900 

0.98 

44.97 

2.75 

89.94 

89.98 

15  !  93.00 

150| 

255} 

446 

560.23 

2.74837 

1.06 

46.47 

2.99 

92.93 

92.98 

30  !  96.00 

156* 

306 

502* 

547.01 

2.73800 

1.14 

47.96 

3.25 

95.92 

95.97 

45  ',  99.00 

202*    316! 

519} 

534.41 

2.72787 

1.23 

49.46 

3.53 

98.90 

98.96 

1100   102.00 

208f 

327f 

536* 

522.39 

2.71799 

1.32 

50.95 

3.82 

101.89 

101.96 

15   105.00 

215 

339f 

554} 

510.92 

2.70835 

1.42 

52.45 

4.12 

104.87 

104.95 

30  108.00 

221* 

351 

612* 

499.95 

2.69893 

1.52 

53.94 

4.44 

107.86 

107.95 

45  111.00 

228 

403} 

631} 

489.46 

2.68972 

1.62 

55.44 

4.78 

110.84 

110.94 

TABLES  99 

TAPERS.    INTERMEDIATE  VALUES  —  (Continued}     A  =  2| 


M 

I 

/ 

b 

„ 

D 

log£> 

d 

t 

X 

V 

Ic 

1200 

114.00 

234! 

415* 

650i 

479.41 

2.68071 

1.73 

56.93 

5.13 

113.82 

113.93 

15 

117.00 

2  41f      4  28i 

710 

469.78 

2.67189 

1.84 

58.42 

5.50 

116.80 

116.92 

30 

120.00 

248! 

44U 

730 

460.55 

2.66328 

1.96 

59.91 

5.88 

119.77 

119.91 

45 

123.00 

256 

454* 

750* 

451.69 

2.65484 

2.08 

61.41 

6.29 

122.75 

122.90 

1300 

126.00 

303J 

508i 

811* 

443.18 

2.64658 

2.21 

62.90 

6.71 

125.72 

125.89 

15   129.00 

310| 

522 

832! 

435.00 

2.63849 

2.34 

64.39 

7.15 

128.69 

128.88 

30 

132.  OOj  3  18* 

536 

754* 

427.14 

2.63057 

2.48 

65.88 

7.61 

131.65 

131.87 

45 

135.00 

326| 

550| 

916! 

419.58 

2.62281 

2.63 

67.37 

8.09 

134.61 

134.86 

1400 

138.00 

334i 

605i 

939* 

412.29 

2.61520 

2.78 

68.86 

8.58 

137.57 

137.84 

15 

141.00 

342i 

620* 

1002! 

405.26 

2.60773 

2.93 

70.34 

9.10 

140.53 

140.82 

30 

144.00 

350* 

636 

1026* 

398.50 

2.60043 

3.09 

71.83 

9.64 

143.48 

143.80 

45 

147.00 

359 

651* 

1050* 

391.98 

2.59326 

3.26 

73.32 

10.19 

146.43 

146.78 

1500 

150.00 

407| 

707* 

11  15 

385.68 

2.58623 

3.43 

74.80 

10.77 

149.37 

149.76 

30 

156.  OOJ  425 

740* 

1205* 

373.72 

2.57255 

3.79 

77.77 

11.99 

155.25 

155.71 

1600 

162.00  443i 

8  14i 

1257* 

362.56 

2.55938 

4.17 

80.73 

13.30 

161.11 

161.65 

30 

168.00 

502 

849* 

1351* 

352.13 

2.54670 

4.58 

83.69 

14.70 

166.94 

167.58 

1700 

174.00 

52H 

926i 

1447* 

342.36 

2.53448 

5.02 

86.64 

16.19 

172.75 

173.50 

30 

180.00 

54U 

1003J 

1545 

333.20 

.52270 

5.48 

89.58 

17.78 

178.54 

179.41 

1800 

186.00 

601! 

1042f 

1644* 

324.61 

.51136 

5.97 

92.52 

19.47 

184.30 

185.31 

30 

192.00 

623 

11  22* 

1745* 

316.53 

.50041 

6.49 

95.46 

21.26 

190.03 

191.20 

1900 

198.00 

644f 

1203! 

1848* 

308.94 

.48987 

7.03 

98.39 

23.15 

195.73 

197.07 

30 

204.00 

707 

1246* 

1953* 

301.78 

.47969 

7.60 

101.31 

25.14 

201.39 

202.93 

2000 

210.00 

730 

1330 

2100 

295.04 

.46988 

8.20 

104.22 

27.25 

207.01 

208.78 

30 

216.00 

753* 

1415 

22  08* 

288.69 

.46043 

8.83 

107.12 

29.47 

212.59 

214.62 

2100 

222.00 

817! 

1500! 

2318* 

282.72 

.45136 

9.50 

110.02 

31.80 

218.13 

220.44 

30 

228.00 

842* 

1548 

2430* 

277.07 

2.44259 

10.19 

112.90 

34.25 

223.62 

226.23 

2200 

234.00 

907! 

1636! 

2544* 

271.75 

2.43417 

10.92 

115.77 

36.81 

229.05 

231.99 

30 

240.00 

933! 

1726J 

2700 

266.74 

2.42609 

11.69 

118.63 

39.49 

234.42 

237.72 

100 


THE  RAILROAD  TAPER 


TAPER  3|.     CURVATURE  INCREASING  3°  00' 
EVERY  30  FT. 


Deflections  to 

at 

B.C. 

C.C. 

C.C. 

C.C. 

C.C. 

C.C. 

C.C. 

C.C. 

3°  00' 

6°  00' 

9°  00' 

12°  00' 

15°  00' 

18°  00' 

21°  00' 

24°  00' 

I 

0 

30 

60 

90 

120 

150 

180 

210 

o 

0          / 

027 

1  07! 

206 

O          / 

322| 

457 

649! 

900 

30 

027 

054 

2  01! 

327 

5  10| 

712 

93H 

60 

1  34! 

054 

1  21 

2  55^ 

448 

658| 

927 

90 

3  18 

228! 

121 

148 

349| 

609 

846! 

120 

5  37| 

439 

3  22! 

1  48 

2  15 

443! 

730 

150 

833 

725! 

600 

4164 

215 

242 

537! 

180 

1204| 

1048 

913| 

721 

5  10| 

242 

309 

210 

16  12 

1446| 

1303 

11  OH 

842 

604! 

309 

M 

l 

0 

D 

log/) 

d 

t 

X 

V 

lc 

600 

30 

054 

955.16 

2.98008 

0.12 

15.  OC 

)   0.24 

30.00 

30.00 

900 

60 

242 

637.25 

2.80431 

0.47 

29.  9^ 

1    1.18 

59.98 

60.00 

1200 

90 

524 

478.84 

2.68021 

1.18 

44.  9f 

»   3.30 

89.91 

89.97 

1500 

120 

900 

384.60 

2.58501 

2.35 

59.  8£ 

>   7.06 

119.67 

119.88 

1800 

150 

1330 

322.75 

2.50887 

4.11 

74.71 

12.91 

149.10 

149.65 

2100 

180 

1854 

279.78 

2.44682 

6.56 

89.4021.29 

177.91 

179.17 

2400 

210 

25  12 

248.99 

2.39618 

9.82 

103.8* 

»32.58 

205.71 

208.28 

TABLES  101 

TAPER  3i     INTERMEDIATE  VALUES'        A- 3 


M 

I 

/ 

b 

0 

D 

logZ) 

d 

t 

X 

V 

Ic 

600 

30.00 

027 

027 

054 

955.16 

2.98008 

0.12 

15.00 

0.24 

30.00 

30.00 

15 

32.50 

029| 

0311 

101 

916.98 

2.96236 

0.14 

16.26 

0.28 

32.50 

32.50 

30 

35.00 

032| 

035! 

108* 

881.75 

2.94535 

0.16 

17.50 

0.33 

35.00 

35.00 

45 

37.50 

035* 

040* 

1  16 

849.13 

2.92897 

0.18 

18.75 

0.39 

37.50 

37.50 

700 

40.00 

038* 

045* 

124 

818.84 

2.91320 

0.20 

20.00 

0.45 

40.00 

40.00 

15 

42.50 

0  41| 

0  50! 

1  32* 

790.65 

2.89798 

0.23 

21.25 

0.52 

42.50 

42.50 

30 

45.00 

045 

056* 

141* 

764.34 

2.88329 

0.26 

22.49 

0.59 

45.00 

45.00 

45 

47.50 

048* 

102 

150* 

739.73 

2.86907 

0.29 

23.74 

0.67 

47.49 

47.50 

800 

50.00 

052 

108 

200 

716.66 

2.85531 

0.32 

24.99 

0.76 

49.99 

50.00 

15 

52.50 

055f 

114| 

210 

694.99 

2.84198 

0.35 

26.24 

0.85 

52.49 

52.50 

30 

55.00 

059| 

120f 

220* 

674.61 

2.82905 

0.39 

27.49 

0.95 

54.99 

55.00 

45 

57.50 

103| 

127* 

231 

655.40 

2.81651 

0.43 

28.74 

1.06 

57.49 

57.50 

900 

60.00 

107* 

134* 

242 

637.25 

2.80431 

0.47 

29.99 

1.18 

59.98 

60.00 

15 

62.50 

1  11| 

1  411 

253* 

620.09 

2.79245 

0.51 

31.24 

1.31 

62.48 

62.50 

30 

65.00 

1  16 

149| 

305* 

603.85 

2.78093 

0.56 

32.48 

1.44 

64.98 

.64.99 

45 

67.50 

2()| 

157 

317* 

588.44 

2.76970 

0.61 

33.73 

1.58 

67.48 

67.49 

1000 

70.00 

25 

205 

330 

573.80 

2.75876 

0.66 

34.98 

1.73 

69.97 

69.99 

15 

72.50 

29| 

213| 

343 

559.88 

2.74810 

0.71 

36.23 

1.89 

72.47 

72.49 

30 

75.00 

34* 

221! 

356* 

546.64 

2.73770 

0.77 

37.48 

2.06 

74.96 

74.99 

45 

77.50 

39* 

230* 

4  10 

534.01 

2.72755 

0.83 

38.73 

2.24 

77.45 

77.49 

1  1  00 

80.00 

44* 

239* 

424 

521.96 

2.71764 

0.89 

39.97 

2.43 

79.95 

79.98 

15 

82.50 

149f 

248! 

438* 

510.46 

2.70796 

0.96 

41.22 

2.63 

82.44 

82.48 

30 

85.00 

155 

258* 

453* 

499.46 

2.69850 

1.03 

42.46 

2.84 

84.93 

84.98 

45 

87.50 

200* 

308 

508* 

488.94 

2.68926 

1.10 

43.71 

3.06 

87.42 

87.47 

1200 

90.00 

206 

318 

524 

478.86 

2.68021 

1.18 

44.96 

3.29 

89.91 

89.97 

15 

92.50 

211! 

3281 

540 

469.20 

2.67136 

1.26 

46.20 

3.54 

92.40 

92.47 

30 

95.00 

217* 

338! 

556* 

459.93 

2.66269 

1.34 

47.45 

3.80 

94.89 

94.96 

45 

97.50 

223| 

349* 

613 

451.03 

2.65421 

1.42 

48.69 

4.07 

97.37 

97.46 

102  THE  RAILROAD  TAPER 

TAPfiR  3|.     INTERMEDIATE  VALUES  —  (Continued)    A  = 


M 

I 

/ 

ft 

0 

D 

log-D 

d 

t 

X 

V 

Ic 

1300 

100.00 

229* 

400* 

630 

442.48 

2.64589 

1.51 

49.94 

4.35 

99.86 

99.95 

15 

102.50 

235f 

411f 

647J 

434.26 

2.63775 

1.60 

51.18 

4.64 

102.35 

102.45 

30 

105.00 

242 

423| 

705* 

426.36 

2.62978 

1.70 

52.42 

4.94 

104.82 

104.94 

45 

107.50 

248i 

435 

723| 

418.752.62195 

1.80 

53.67 

5.26 

107.30 

107.43 

1400 

110.00 

255 

447 

742 

411.41 

2.61427 

1.90 

54.91 

5.59 

109.78 

109.92 

15 

112.50 

301f 

459* 

801 

404.34 

2.60675 

2.01 

56.15 

5.94 

112.26 

112.41 

30 

115.00 

308| 

511! 

820* 

397.53 

2.59937 

2.12 

57.39 

6.30 

114.73 

114.90 

45 

117.50 

315| 

524| 

840 

390.95 

2.59212 

2.23 

58.64 

6.67 

117.20 

117.39 

1500 

120.00 

322| 

537| 

900 

384.60 

2.58501 

2.35 

59.88 

7.06 

119.67 

119.88 

30 

125.00 

337 

604* 

941* 

372.53 

2.57116 

2.60 

62.36 

7.88 

124.61 

124.85 

1600 

130.00 

352 

632 

1024 

361.26 

2.55782 

2.87 

64.84 

8.76 

129.54 

129.82 

30 

135.00 

407£ 

700! 

1108* 

350.70 

2.54494 

3.15 

67.31 

9.70 

134.45 

134.79 

1700 

140.00 

423£ 

730| 

1154 

340.79 

2.53249 

3.45 

69.78 

10.70 

139.34 

139.75 

30 

145.00 

440 

801J 

1241* 

331.49 

2.52047 

3.77 

72.25 

11.77 

144.22 

144.70 

180Q 

150.00 

457 

833 

1330 

322.75 

2.50887 

4.11 

74.71 

12.91 

149.10  149.65 

30 

155.00 

514| 

905! 

1420* 

314.51 

2.49763 

4.47 

77.17 

14.12 

153.96 

154.59 

1900 

160.00 

532J 

939| 

1512 

306.75 

2.48678 

4.84 

79.63 

15.40 

158.78 

159.52 

30 

165.00 

551 

1014* 

1605* 

299.42 

2.47628 

5.24 

82.08 

16.76 

163.59 

164.44 

2000 

170.00 

610 

1050 

1700 

292.50 

2.46613 

5.66 

84.52 

18.19 

168.39 

169.36 

30 

175.00 

629J 

1126! 

17  56* 

285.96 

2.45631 

6.10 

86.96 

19.70 

173.16 

174.27 

2100 

180.00 

649J 

12041 

1854 

279.78 

2.44682 

6.56 

89.40 

21.29 

177.91 

179.17 

30 

185.00 

710 

12  434l 

1953* 

273.92 

2.43762 

7.04 

91.83 

22.96 

182.62 

184.06 

2200 

190.00 

731 

1323 

2054 

268.382.42875 

7.55 

94.25 

24.71 

187.30 

188.93 

30 

195.00 

752* 

1403f 

21  56* 

263.132.42017    8.08 

96.67 

26.55 

191.95 

193.79 

2300 

200.00 

814| 

1445£ 

2300 

258.172.41191    8.64 

99.08 

28.47 

196.59 

198.63 

2400 

210.00  900 

1612 

25  12 

248.99 

2-.39618    9.82 

103.88 

32.58 

205.71 

208.28 

1 

TABLES 


103 


TAPER  3*.    CURVATURE  INCREASING  4°  00' 
EVERY  30  FT. 


Deflections  to 

at 

B.C. 

C.C. 

C.C. 

C.C. 

C.C. 

C.C. 

C.C. 

4°  00' 

8°  00' 

12°  00' 

16°  00' 

20°  00' 

24°  00' 

28°  00' 

I 

0 

30 

60 

90 

120 

150 

180 

0 

0          / 

036 

1  30 

2  48 

430 

636 

906 

30 

036 

1  12 

242 

436 

6  54 

936 

60 

206 

1  12 

1  48 

3  54 

624 

9  18 

90 

424 

3  18 

1  48 

224 

506 

8  12 

120 

730 

612 

430 

224 

300 

618 

150 

1124 

954 

800 

542 

300 

336 

180 

1606 

1424 

1218 

948 

654 

336 

M 

l 

0 

D 

logD 

d 

t 

X 

I 

lc 

800 

30 

1  12 

716.50 

2.85522 

0.16 

15.00 

0.31 

30.  OC 

30.00 

1200 

60 

336 

478.31 

2.67971 

0.63 

29.98 

1.57 

59.97 

59.99 

1600 

90 

712 

359.96 

2.55625 

1.57 

44.92 

4.39 

89.84 

89.95 

2000 

120 

1200 

289.97 

2.46235 

3.13 

59.78 

9.40 

119.42 

119.79 

2400 

150 

1800 

244.63 

2.38851 

5.46 

74.49 

17.17 

148.  4C 

149.39 

2800 

180 

25  12 

213.85 

2.33011 

8.71 

88.95 

28.24 

176.28 

178.52 

104  THE  KAILROAD  TAPER 

TAPER  3£.  INTERMEDIATE  ^VALUES     A  =  4 


M 

I 

/ 

b 

o 

D 

*D 

d 

t 

X 

V 

Ic 

800 

30.00 

036 

036 

12 

716.50 

2.85522 

0.16 

15.00 

0.31 

30.00 

30.00 

15 

31.88 

038! 

040 

181 

694.82 

2.84187 

0.18 

15.93 

0.36 

31.87 

31.87 

30 

33.75 

041$ 

044$ 

26 

674.42 

2.82893 

0.20 

16.87 

0.41 

33.74 

33.75 

45 

35.62 

0441 

0491 

33$ 

655.19 

2.81637    0.22 

17.81 

0.46 

35.62 

35.62 

900 

37.50 

0471 

054 

411 

637.02 

2.80415    0.24 

18.74 

0.52 

37.50 

37.50 

15 

39.38 

0501 

059 

491 

619.84 

2.79228 

0.26 

19.68 

0.58 

39.37 

39.37 

30 

41.25 

053$ 

104 

1  57$ 

603.58 

2.78073 

0.29 

20.62 

0.64 

41.24 

41.25 

45 

43.12 

056! 

1  09$ 

2061 

588.15 

2.76949    0.32 

21.55 

0.71      43.12 

43.12 

1000 

45.00 

100 

1  15 

2  15 

573.49 

2.75853 

0.35 

22.49 

0.79      44.99 

45.00 

15 

46.88 

103$ 

1  20! 

2241 

559.55 

2.74784 

0.38 

23.43 

0.87      46.87 

46.87 

30 

48.75 

107 

1  26| 

233$ 

546.28 

2.73742 

0.41 

24.36 

0.95     48.74    48.75 

45 

50.62 

110$ 

132f 

2431 

533.62 

2.72723    0.44 

25.30 

.04      50.61 

50.62 

1100 

52.50 

141 

139 

2531 

521.54 

2.71729    0.47 

26.24 

.13      52.48 

52.50 

15 

54.38 

18 

145$ 

303$ 

510.01 

2.70758    0.51 

27.17 

.23     54.35    54.37 

30 

56.25 

22 

1  52 

3  14 

498.98 

2.69808    0.55 

28.11 

.34     56.22    56.24 

45 

58.12 

26 

1  58! 

324! 

488.43 

2.68880 

0.59 

29.04 

.45 

58.10 

58.12 

1200 

60.00 

130 

206 

336 

478.31 

2.67971 

0.63 

29.98 

.57 

59.97 

59.99 

15 

61.88 

1341 

2131 

347$ 

468.61 

2.67081 

0.67 

30.91 

.69 

61.84 

61.87 

30 

63.75 

138| 

2  20$ 

359 

459.31 

2.66211 

0.72 

31.85 

.82     63.71    63.74 

45 

65.62 

142! 

2281 

411 

450.38 

2.65358 

0.77 

32.79 

.96 

65.59 

65.62 

1300 

67.50 

1471 

236 

4231 

441.79 

2.64522 

0.82 

33.72 

2.11 

67.46 

67.49 

15 

69.38 

151! 

244 

435! 

433.53 

2.63702 

0.87 

34.65 

2.26 

69.33 

69.36 

30 

71.25 

1  56f 

252 

448$ 

425.58 

2.62898 

0.92 

35.59 

2.42 

71.19 

71.23 

45 

73.12 

2011 

300$ 

501! 

417.92 

2.62109 

0.97 

36.52 

2.58 

73.06 

73.11 

1400 

75.00 

206 

309 

515 

410.54 

2.61336 

1.03 

37.46 

2.75 

74.93 

74.98 

15 

76.88 

2  11 

317! 

528! 

403.42 

2.60576 

.09 

38.39 

2.93 

76.79 

76.85 

30 

78.75 

216 

326$ 

542$ 

396.56 

2.59831 

.15 

39.33 

3.12     78.66 

78.72 

45 

80.62 

221 

335! 

556! 

889.94 

2.59100 

.22 

40.26 

3.31  '  80.53 

80.59 

1500 

82.50 

2  261 

345 

6111 

383.53 

2.58380 

.28 

41.19 

3.51      82.39 

82.48 

30 

86.25 

237 

404 

6*41 

371.35 

2.56978 

.42 

43.06 

3  93     86.12 

86.21 

1600 

90.00 

248 

424 

712 

359.96 

2.55625 

.57 

44.92 

4.39      89.84 

89.95 

30 

93.75 

259$ 

444$ 

744 

349.28 

2.54317 

.73 

46.79 

4.89     93.56 

93.69 

1700 

97.50 

3  111 

506 

8171 

339.23 

2.53049 

.89 

48.65 

5.42      97.27 

97.43 

30 

101.25 

323i 

528 

851$ 

329.79 

2.51824 

2.07 

50.51 

5.98    100.98 

101.16 

1800 

105.00 

336 

551 

927 

320.90 

2.50637 

2.26 

52.37 

6.59  ;104.68 

104.89 

30 

108.75 

349 

614$ 

1003$ 

312.50 

2.49485 

2.46 

54.22 

7.23    108.38  108.62 

1900 

112.50 

4021 

639 

10411 

304.58 

2.48370 

2.67 

56.08 

7.91    112.07 

112.34 

30 

116.25 

416 

704 

1120 

297.07 

2.47286 

2.89 

57.93 

8.63  '115.75 

116.07 

2000 

120.00 

430 

730 

1200 

289.97 

2.46235 

3.13 

59.78 

9.40  1119.42  119.79 

30 

123.75 

444$ 

756$ 

1241 

283.24 

2.45215 

3.38 

61.63 

10.21    123.08 

123.51 

2100 

127.50 

4591 

824 

13231 

276.86 

2.44226 

3.64 

63.48 

11.06  126.74 

127.22 

30 

131.25 

514$ 

852 

1406$ 

270.79 

2.43263 

3.91 

65.32 

11.96    130.38130.93 

TABLES  105 

TAPER  3|.     INTERMEDIATE  VALUES  —  (Continued)    A  =  4 


M 

I 

/ 

b 

0 

D 

logZ) 

d 

t 

X 

V 

Ic 

2200 

135.00 

530 

921 

1451 

265.02 

2.42328 

4.19 

67.16 

12.90 

134.01 

134.63 

30 

138.75 

546 

950£ 

1536£ 

259.53 

2.41419 

4.48 

69.00 

13.90 

137.63 

138.33 

2300 

142.50 

602i 

1021 

1623i 

254.33 

2.40540 

4.80 

70.83 

14.94 

141.23 

142.02 

2400 

150.00 

636 

1124 

1800 

244.63 

2.38851 

5.46 

74.49 

17.17 

148.40 

149.39 

2500 

157.50 

7  llj 

1230 

19  4U 

235.82 

2.37258 

6.18 

78.14 

19.61 

155.50 

156.73 

2600 

165.00 

748 

1339 

2127 

227.81 

2.35757 

6.97 

81.76 

22.26 

162.52 

164.04 

2700 

172.50 

826^ 

1451 

23  m 

220.51 

2.34343 

7.81 

85.37 

25.14 

169.45 

171.31 

2800 

180.00 

906 

1606 

2512 

213.85 

2.33011 

8.71 

88.95 

28.24 

176.29 

178.52 

TAPER  3f.     CURVATURE   INCREASING  5°  00' 

EVERY  30  FT.  A  =  5 


Transit 
at 

Deflections  to 

B.C. 

5°  00' 

C.C. 

10°  00' 

C.C. 

15°  00' 

C.C. 

20°  00' 

C.C. 

25°  00' 

C.C. 

30°  00' 

I 

0 

30 

60 

90 

120 

150 

0 
30 
60 
90 
120 
150 

045 
237£ 
530 
922^ 
1415 

045 

"l30' 
407£ 
745 
12  22| 

0           / 

1521 
130 

0              / 

330 
322| 
215 

"300" 
707£ 

537£ 
545 
452i 
300 

'345" 

8  15 
837£ 
800 
622i 
345 

215 

537^ 
1000 

M 

I                 0 

D 

log£> 

d 

t 

X 

y 

lc 

•    / 
1000 
1500 
2000 
2500 
3000 

30       130 
60      430 
90      900 
120     15  00 
150     22  30 

573.34 
383.13 
288.80 
233.54 
198.34 

2.75841 
2.58323 
2.45939 
2.36836 
2.29741 

0.20 

0.78 
1.96 
3.90 
6.81 

14.99 
29.97 
44.88 
59.66 
74.21 

0.39 
1.96 
5.49 
11.73 
21.89 

30.00 
59.96 
89.75 
119.09 
147.50 

30.00 
59.99 
89.92 
119.67 
149.04 

106 


THE  RAILROAD  TAPER 


TAPER  3|.     INTERMEDIATE  VALUES 


M 

I 

/ 

b 

0 

D 

logD 

d 

t 

X 

f 

Ic 

1000 

30.00 

045 

0       / 

045 

1  30 

573.34 

2.75841 

0.20 

14.99 

0.39 

30.00 

30.00 

15 

31.50 

0471 

049 

136| 

559.38 

2.74771 

0.21 

15.74 

0.43 

31.50 

31.50 

30 

33.00 

050} 

053! 

144 

546.10 

2.73727 

0.23 

16.49 

0.48 

33.00 

33.00 

45 

34.50 

053} 

058 

151} 

533.43 

2.72708 

0.25 

17.24 

0.53 

34.50 

34.50 

1100 

36.00 

056 

102| 

158f 

521.34 

2.71712 

0.27 

17.99 

0.59 

36.00 

36.00 

15 

37.50 

059} 

107} 

206* 

509.79 

2.70739 

0.29 

18.74 

0.65 

37.49 

37.50 

30 

39.00 

102 

1  12* 

214* 

498.74 

2.69787 

0.31 

19.49 

0.71 

38.99 

39.00 

45 

40.50 

105} 

117* 

222! 

488.18 

2.68858 

0.34 

20.24 

0.77 

40.49 

40.50 

1200 

42.00 

108* 

1  221 

231} 

478.05 

2.67947 

0.37 

20.99 

0.84 

41.99 

42.00 

15 

43.50 

1  111 

1  28 

2391 

468.34 

2.67056 

0.40 

21.74 

0.91 

43.49 

43.50 

30 

45.00 

1  15 

133| 

2481 

459.02 

2.66183 

0.43 

22.49 

0.98 

44.99 

45.00 

45 

46.50 

1  18} 

139} 

2571 

450.07 

2.65328 

0.46 

23.23 

.06 

46.49 

46.50 

1300 

48.00 

122 

145} 

307} 

441.46 

2.64489 

0.49 

23.98 

.14 

47.98 

48.00 

15 

49.50 

125* 

151} 

3161 

433.18 

2.63667 

0.52 

24.73 

.23 

49.48 

49.50 

30 

51.00 

129 

157* 

326* 

425.21 

2.62860 

0.55 

25.48 

.32 

50.98 

51.00 

45 

52.50 

132f 

2  03  f 

336* 

417.53 

2.62069 

0.58 

26.23 

.42 

52.48 

52.50 

1400 

54.00 

136| 

210} 

3461 

410.13 

2.61292 

0.62 

26.98 

1.52 

53.97 

53.99 

15 

55.50 

1  40* 

2161 

357} 

402.99 

2.60529 

0.66 

27.72 

62 

55.47 

55.49 

30 

57.00 

144} 

223f 

408 

396.11 

2.59782 

0.70 

28.47 

.73 

56.97 

56.99 

45 

58.50 

148} 

230* 

4181 

389.46 

2.59046 

0.74 

29.22 

1.84 

58.46 

58.49 

1500 

60.00 

152* 

237* 

430 

383.03 

2.58323 

0.78 

29.97 

1.96 

59.96 

59.99 

30 

63.00 

201 

252 

453 

370.80 

2.56914 

0.87 

31.46 

2.21 

62.95 

62.99 

TABLES  107 

TAPER  3|.     INTERMEDIATE  VALUES  -  (Continued)    A  =  5 


M 

I 

/ 

b 

0 

D 

logD 

d 

t 

X 

y 

Ic 

1600 

66.00 

209| 

307 

0        1 

5161 

359.35 

2.55552 

0.96 

32.96 

2.48 

64.44 

64.99 

30 

69.00 

218* 

323 

541* 

348.61 

2.54234 

1.06 

34.45 

2.77 

68.92 

68.98 

1700 

72.00 

227f 

339* 

6071 

338.51 

2.52957 

1.17 

35.94 

3.09 

71.90 

71.97 

30 

75.00 

237* 

356i 

6331 

329.00 

2.51720 

.28 

37.44 

3.43 

74.88 

74.97 

1800 

78.00 

247* 

4  131 

701i 

320.04 

2.50520 

.40 

38.92 

3.79 

77.86 

77.96 

30 

81.00 

257* 

432 

729* 

311.57 

2.49356 

.53 

40.41 

4.18 

80.84 

80.95 

1900 

84.00 

308 

4-501 

7581 

303.58 

2.48227 

.67 

41.90 

4.59 

83.81 

83.94 

30 

87.00 

319 

510 

829 

295.99 

2.47128 

.81 

43.39 

5.03 

86.78 

86.93 

2000 

90.00 

330 

530 

900 

288.80 

2.46060 

.96 

44.88 

5.49 

89.75 

89.92 

30 

93.00 

341* 

550| 

932 

281.98 

2.45022 

2.12 

46.36 

5.98 

92.71 

92.91 

2100 

96.00 

353 

611f 

10  04J 

275.50 

2.44012 

2.28 

47.85 

6.50 

95.67 

95.89 

30 

99.00 

405 

633* 

1038* 

269.33 

2.43028 

2.45 

49.33 

7.05 

98.62 

98.87 

2200 

102.00 

417* 

6551 

11  13J 

263.46 

2.42071 

2.63 

50.81 

7.63 

101.56 

101.85 

30 

105.00 

430 

7  181 

11481 

257.872.41140 

2.82 

52.29 

8.23 

104.50 

104.82 

2300 

108.00 

443 

742J 

12251 

252.55 

2.40235 

3.02 

53.76 

8.86 

107.44 

107.78 

2400 

114.00 

509* 

83U 

13401 

242.61 

2.38491 

3.44 

56.72 

10.23 

113.28 

113.73 

2500 

120.00 

537* 

922* 

1500 

233.54 

2.36836 

3.90 

59.66 

11.73 

119.09 

119.67 

2600 

126.00 

606* 

1016J 

16221 

225.24 

2.35265 

4.40 

62.59 

13.37 

124.87 

125.59 

2700 

132.00 

637 

11  121 

17491 

217.64 

2.33774 

4.94 

65.51 

15.15  130.61 

131.49 

2800 

138.00 

708* 

12  10f 

19  191 

210.66 

2.32358 

5.52 

68.42 

17.07 

136.29 

137.36 

2900 

144.00 

741 

13111 

20521 

204.24 

2.31014 

6.14 

71.32 

19.15 

141.93 

143.21 

3000 

150.00 

815 

14  15 

2230 

198.34 

2.29741 

6.81 

74.21 

21.39 

147.50 

149.04 

108 


THE   RAILROAD  TAPER 


* 

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.    •    .       .    .    .       .    .   .                     .    .    .       .    .    . 

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TABLES 


109 


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o  lO        o 

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0 


880    8  SH 


«s  safe 

>  Oo  «OOT~t        QC'o 

I          O          O          o  C^  r_|          CO          O 


*    883    888s 


OOo 


&    SSff 

C^        O  (M  O 


•0 


•~»>'^O         ^-~>^O         f-o"QO         i^i^O 


O        "-^  'tS   O 


8 


8 


(N 


110 


THE   RAILROAD  TAPER 


$ 


ft 


^ 


Oo 


52* 


CO        o 
0 


23    82~    82g    S8g 


82 


r  H°r 


:*    893- 


<N  O 


OO10       OOo          OOo          OO 
*^o         C^       cj       CO       (>!       ^r 


I  ISs  15? 


<N        ^        CO 


1  «i-      Q  <£>  X. 
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•~^"t3O        ~~s^Ci        •^"•eo 


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TABLES 


111 


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CO 


O  ^  O  C3  ^          O^t1^ 

Oo  COOo  cO  O  *^ 

O 


OS  *"  O  ^f  ' 

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Sfe 


5? 


88$      8c^H; 

OOo          iO  O  "• 

l^  T-H  *>          O 


8°  I,    8 


00- 


Oo 
c<l 


B 


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<N        cO^O 


8^0    35%    8S§ 


8iO^ 
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10  rH' 

CO        o 

"-1        CO 


83 


88;? 


'<Mo          O  iM 


ocoo 


CO          OS 


O    •    —  "SO        -o"Q   O        "~i"?3O 


O        i»o  'XJ   O 


8 


^          8 


112 


THE    RAILROAD  TAPER 


,-H  CO  O 


883 


Oo 


LO  O  ^ 

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88.8 

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ScO^        ^^So*       P^c 


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(M         «O 


22         •    -S 
O       nnoco 


C<i        co       o          f—       o 
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000 


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l>        o  OO        o 

(M  CO 


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O  Oo 


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i— i  o     O'*1^     o t"- rs     <~>| 

T— i  o  Or- <0  OI-HO  O1 


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838 

O-^o 


og    8Ssg,    88g| 


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OCOo 


OOo 


8^^    8^: 


OOo  OI-HO 


m      co      co 


888a    81 


O  OOo          OOo 

S      co      22  ^  ^ 

CQ          ,-^          *^          ^- 


8 


8 


8 


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TABLES 


St">-  *" 
co^11 


113 

33?   SBfe    889 


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S?    88s 


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8  §3$-    8£ 

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'o"        8^0°° 
00        ^        O5 


80  ^       o  • 
S^    Si 

8^:r  s1 


ssr- 

10  CO  ^ 


to  10  w      o  co  i 


O  i-l  ^H  i-H 


8Oi^ 
.^^5 

OCOo 

S    2 


88^3-    8feo    8S« 

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00 


r^^O         r^»^O         p-o'-^O         p^^^O         f-^^O         F^o^O 


8          8          8          8          8          8 


114 


THE   RAILROAD  TAPER 


«5 

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00  i~H  ^^        ^^  I-H  o           to  C^l  ^^        ^^  CO  o           ^^  ^O  o 
i>-o           OiojOQ           C^^^fN 

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^?  T—  *  ^.         <O  CO  ^p        ^D  OO  *^               •                         ... 

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O^o           O»OI>        4O0010 

^    ^         ^    s     •          •  •  • 

ST 

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2        ^        £J        ?Q        ^^C^ 

1 

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<N 

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C« 

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O  Oi  o           G  C^  o 
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s 

ZZs 

H« 

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\ 

1—4 

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tl 

^ 

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22          S          8^          8          S 

TABLES 

ALIGNMENT  REVISION 
MAINTAINING  LENGTH  UNCHANGED 

(See  page  55,  par.  43) 


115 


M 

r 

00' 

1°30' 

2°  00' 

A 

1 

1 

1 

i 

Taper. 

i 

* 

* 

1 

I 

I 

h 

M' 

I 

h 

M' 

I 

h 

M' 

I 

h 

M' 

2°  30' 
5 
10 

15 
20 
25 

110.00 
96.00 

0.04 
0.14 

1°03' 

m.  60 

158.00 
154.00 

0.01 
0.23 

0.35 

1°48' 
1°34' 

1°32' 

266!6o 

224.00 
218.00 
214.00 

•QA2 

0.37 
0.50 
0.26 

2°  25' 

2°  07' 
2°  04' 

2°  02' 

110.00 
94.00 

92.00 

0.07 
0.22 

0.19 

2°  20' 
2°  04' 

2°  02' 

M 

2°  30' 

3°  00' 

A 

1 

i 

i 

! 

Taper. 

i 

i 

1 

H 

' 

I 

k 

M' 

i 

h 

M' 

I 

h 

M' 

I 

h 

0.03 

0.12 
0.17 

M' 

10° 

15 
20 
25 

30 
40 
50 

350.00 
296.00 
286.00 

280.00 
276.00 
274.00 

0.84 
0.71 
0.87 

0.75 
1.05 
1.25 

3°  10' 
2°  43' 
2°  38' 

2°  35' 
2°  33' 
2°  32' 

132.00 

125.00 
123.00 

0.15 

0.19 
0.27 

2°  42' 

2°  35' 
2°  33' 

165.00 
157.00 
154.00 

153.00 

0.31 
0.32 
0.25 

0.35 

3°  14' 
3°  07' 
3°  04' 

3°  03' 

96.67 

93.33 
92.00 

3°  10' 

3°  05' 
3°  03' 

M 

4°  00' 

A 

f 

i 

> 

Taper. 

1 

H 

2 

I 

I 

h 

M' 

I             h            M' 

I            h            M' 

10° 
15 
20 
25 

30 
40 
50 
60 

i56!66      '6.34      4°  30' 
139.33       0.35       4°  14' 
135.33       0.31       4°  08' 

133.33       0.24       4°  05' 
132.00       0.35      4°  03' 

110.00       0.15      4°  40' 
96.00       0.12      4°  12' 
93.00       0.10      4°  06' 
92.00       0.15       4°  04' 

91.50       0.20      4°  03' 

259!  66 

233.00 

224.00 
217.00 
214.00 
213.00 

0.81 

0.76 
0.69 
0.51 
0.75 

4°  49' 
4°  23' 

4°  14' 
4°  07' 
4°  04' 
4°  03' 



116 


THE    RAILROAD    TAPER 


MAINTAINING   LENGTH   UNCHANGED  —  (Continued} 


M 

5°  00' 

A 

1 

1 

1 

If 

Taper. 

1 

| 

2 

21 

I 

I 

h 

M' 

I 

h 

AT 

I 

h  . 

M' 

I 

i 

M' 

5°  30' 
5°  13' 
5°  08' 
5°  05' 
5°  03' 

15° 
20 
25 
30 
40 
50 
60 
70 
80 
90 

154.00 
132.50 
127.00 
124.50 
122.50 
121.50 

0.35 
0.34 
0.32 
0.29 
0.32 
0.28 

6°  08' 
5°  25' 
5°  14' 
5°  09' 
5°  05' 
5°  03' 

102.00 
95.20 
93.20 
92.00 
91.20 

0.25 
0.17 
0.18 
0.13 
0.19 

6°  19' 
5°  26' 
5°  15' 
5°  10' 
5°  07' 
5°  05' 
5°  04' 

223.33 
190.00 
182.67 
176.67 
174.00 
172.67 
172.00 

0.76 
0.58 
0.64 
0.70 
0.66 
0.62 
0.72 

6°  20' 
5°  30' 
5°  19' 
5°  10' 
5°  06' 
5°  04' 
5°  03' 

349.00 
296.00 
285.00 
280.00 
277.00 
275.00 
274.00 

1.64 
1.43 
1.50 
1.60 
1.53 
1.31 
1.48 

M 

6°  00' 

A 

f 

1 

li 

1^ 

Taper. 

5 

2 

2i 

2* 

I 

z 

h 

M' 

I 

h 

M' 

I 

h 

M' 

I 

h 

M' 

15° 

20 
25 
30 
40 
50 
60 
70 
80 
90 

0^63 
0.58 
0.66 
0.52 
0.76 
0.58 

111.00 
98.33 
95.00 
93.33 
92.00 
91.00 

0.25 
0.22 
0.24 
0.25 
0.34 
0.14 

7°  03' 
6°  25' 
6°  15' 
6°  10' 
6°  06' 
6°  03' 

6^47' 
6°  27' 
6°  14' 
6°  08' 
6°  06' 
6°  04' 

131.60 
123.60 
120.00 
117.20 
116.00 
115.20 

0.35 
0.36 
0.33 
0.35 
0.34 
0.20 

6°  44' 
6°  24' 
6°  15' 
6°  08' 
6°  05' 
6°  03' 

258.67 
228.67 
220.67 
217.33 
215.33 
214.00 
212.67 

1.21 
1.12 
1.09 
1.25 
1.35 
1.39 
0.72 

7°  13' 
6°  28' 
6°  16' 
6°  11' 
6°  08' 
6°  06' 
6°  04' 

173.50 
163.50 
157.00 
154.00 
153.00 
152.00 

::::  ::::: 

5 

8°  00' 

A 

1 

H 

H 

-    2 

Taper. 

2 

2J 

2* 

2! 

/ 

I 

h 

M' 

I 

h 

M' 

I 

h 

M' 

I 

h 

0.43 
0.40 
0.43 
0.31 
0.31 
0.43 
0.33 

M' 

20° 
25 
30 
40 
50 
60 
70 
80 
90 

112.25 
101.00 
97.00 
93.25 
92.00 
91.50 
91.00 

9°  29' 
8°  44' 
8°  28' 
8°  13' 
8°  08' 
8°  06' 
8°  04' 

176.67 
150.00 
139.00 
135.33 
133.67 
132.67 
132.00 
131.67 

0.71 
0.67 
0.64 
0.63 
0.70 
0.75 
0.74 
0.95 

10°  20' 
9    00' 
8    27' 
8    16' 
8    11' 
8    08' 
8    06' 
8    05' 

258  '50 
232.50 
224.00 
219.50 
217.00 
215.50 

!ei 

.58 
.57 
.48 
.50 
.65 

9d  37' 
8°  45' 
8°  28' 
8°  19' 
8°  14' 
8°  11' 

220.40 
179.60 
172.00 
168.40 
166.40 
165.60 
164.80 

1.04 
0.96 
1.08 
0.92 
0.81 
1.14 
1.18 

10  26' 
8  44' 
8  25' 
8  16' 
8  11' 
8  09' 
8  07' 

M 

10°  00' 

A 

U 

ij 

2 

2^ 

Taper. 

2i 

2* 

2| 

3 

I 

I 

h 

M' 

J 

h 

A/' 

I 

h 

M' 

I 

h 

M' 

25° 
30 
40 
50 
60 
70 
80 
90 

258  '.00 
234.80 
226.40 
221.60 
218.80 

'6!76 
0.69 
0.65 
0.73 
0.64 
0.71 
0.82 

12°  22' 
10°  50' 
10°  28' 
10°  19' 
10°  13' 
10°  10' 
10°08' 

112.00 
101.40 
95.40 
93.40 
92.20 
91.60 
91.20 
91.00 

0.45 
0.41 
0.39 
0.45 
0.39 
0.42 
0.41 
0.53 

11°50' 
10°57' 
10°27' 
10°  17' 

io°ir 

10°08' 
10°06' 
10°  05' 

L99 
1.90 
1.95 
1.86 
1.90 

12°  bb' 
iro2 

10°  41 
10°  29 
10°  22 

2i9!6o 

190.67 
182.67 
178.67 
176.33 
175.00 

^36 
.32 
.33 
.31 
.30 
1.46 

12°  27' 
11°  02' 
10°  38' 
10°  26' 
10°  19' 
10°  15 

155.50 
132.50 
127.00 
124.75 
123.25 
122.50 
122.  OC 

LOGARITHMS  OF  NUMBERS  117 


100 

1 
2 
3 
4 
5 
6 
7 
8 
9 

110 

1 


4 
5 
6 
7 
8 
9 

120 

1 
2 
3 
4 
5 
6 
7 
8 
9 

130 

1 
2 
3 
4 
5 
6 
7 
8 
9 

140 

1 

2 
3 
4 
5 
6 
7 
8 
9 

150 


3456789 


00000  00043  00087  00130  00173  00217  00260  00303  00346  00389 

0432  0475  0518  0561  0604  0647  0689  0732  0775  0817 

0860  0903  0945  0988  1030  1072  1115  1157  1199  1242 

1284  1326  1368  1410  1452  1494  1536  1578  1620  1662 

1703  1745  1787  1828  1870  1912  1953  1995  2036  2078 

2119  2160  2202  2243  2284  2325  2366  2407  2449  2490 

2531  2572  2612  2653  2694  2735  2776  2816  2857  2898 

2938  2979  3019  3060  3100  3141  3181  3222  3262  3302 

3342  3383  3423  3463  3503  3543  3583  3623  3663  3703 

3743  3782  3822  3862  3902  3941  3981  4021  4060  4100 

04139  04179  04218  04258  04297  04336  04376  04415,  04454  04493 

4532  4571  4610  4650  4689  4727  4766  4805  4844  4883 

4922  4961  4999  5038  5077  5115  5154  5192  5231  5269 

5346  5385  5423  5461  5500  5538  5576  5614  5652 


5690  5729  5767  5805  5843  5881  5918  5956  5994  6032 

6070  6108  6145  6183  6221  6258  6296  6333  6371  6408 

6446  6483  6521  6558  6595  6633  6670  6707  6744  6781 

6819  6856  6893  6930  6967  7004  7041  7078  7115  7151 

7188  7225  7262  7298  7335  7372  7408  7445  7482  7518 

7555  7591  7628  7664  7700  7737  7773  7809  7846  7882 

07918  07954  07990  08027  08063  08099  08135  08171  08207  08243 

8279  8314  8350  8386  8422  8458  8493  8529  8565  8600 

8636  8672  8707  8743  8778  8814  8849  8884  8920  8955 

8991  9026  9061  9096  9132  9167  9202  9237  9272  9307 

9342  9377  9412  9447  9482  9517  9552  9587  9621  9656 

9691  9726  9760  9795  9830  9864  9899  9934  9968  10003 

10037  10072  10106  10140  10175  10209  10243  10278  10312  0346 

0380  0415  0449  0483  0517  0551  0585  0619  0653  0687 

0721  0755  0789  0823  0857  0890  0924  0958  0992  1025 

1059  1093  1126  1160  1193  1227  1261  1294  1327  1361 

11394  11428  11461  11494  11528  11561  11594  11628  11661  11694 

1727  1760  1793  1826  1860  1893  1926  1959  1992  2024 

2057  2090  2123  2156  2189  2222  2254  2287  2320  2352 

2385  2418  2450  2483  2516  2548  2581  2613  2646  2678 

2710  2743  2775  2808  2840  2872  2905  2937  2969  3001 

3033  3066  3098  3130  3162  3194  3226  3258  3290  3322 

3354  3386  3418  3450  3481  3513  3545  3577  3609  3640 

S672  3704  3735  3767  3799  3830  3862  3893  3925  3956 

3988  4019  4051  4082  4114  4145  4176  4208  4239  4270 

4301  4333  4364  4395  4426  4457  4489  4520  4551  4582 

14613  14644  14675  14706  14737  14768  14799  14829  14860  14891 

4922  4953  4983  5014  5045  5076  5106  5137  5168  5198 

5229  5259  5290  5320  5351  5381  5412  5442  5473  5503 

5534  5564  5594  5625  5655  5685  5715  5746  5776  5806 

5836  5866  5897  5927  5957  5987  6017  6047  6077  6107 

6137  6167  6197  6227  6256  6286  6316  6346  6376  6406 

6435  6465  6495  6524  6554  6584  6613  6643  6673  6702 

6732  6761  6791  6820  6850  6879  6909  6938  6967  6997 

7026  7056  7085  7114  7143  7173  7202  7231  7260  7289 

7319  7348  7377  7406  7435  7464  7493  7522  7551  7580 

17609  17638  17667  17696  17725  17754  17782  17811  17840  17869 


118  LOGARITHMS  OF  NUMBERS 


N 

150 

1 
2 
3 
4 
5 
6 
7 
8 
9 

160 

1 
2 
3 
4 
5 
6 
7 


170 

1 
2 
3 
4 
5 
6 
7 
8 
9 

180 
1 
2 
3 
4 
5 
6 
7 
8 


190 

1 
2 
3 
4 
5 
6 
7 
8 
9 

200 


23456789 


17609  17638  17667  17696  17725  17754  17782  17811  17840  17869 

7898  7926  7955  7984  8013  8041  8070  8099  8127  8156 

8184  8213  8241  8270  8298  8327  8355  8384  8412  8441 

8469  8498  8526  8554  8583  8611  8639  8667  8696  8724 

8752  8780  8808  8837  8865  8893  8921  8949  8977  9005 

9033  9061  9089  9117  9145  9173  9201  9229  9257  9285 

9312  9340  9368  9396  9424  9451  9479  9507  9535  9562 

9590  9618  9645  9673  9700  9728  9756  9783  9811  9838 

9866  9893  9921  9948  9976  20003  20030  20058  20085  20112 

20140  20167  20194  20222  20249  0276  0303  0330  0358  0385 

20412  20439  20466  20493  20520  20548  20575  20602  20629  20656 

0683  0710  0737  0763  0790  0817  0844  0871  0898  0925 

0952  0978  1005  1032  1059  1085  1112  1139  1165  1192 

1219  1245  1272  1299  1325  1352  1378  1405  1431  1458 

1484  1511  1537  1564  1590  1617  1643  1669  1696  1722 

1748  1775  1801  1827  1854  1880  1906  1932  1958  1985 

2011  2037  2063  2089  2115  2141  2167  2194  2220  2246 

2272  2298  2324  2350  2376  2401  2427  2453  2479  2505 

2531  2557  2583  2608  2634  2660  2686  2712  2737  2763 

2789  2814  2840  2866  2891  2917  2943  2968  2994  3019 

23045  23070  23096  23121  23147  23172  23198  23223  23249  23274 

3300  3325  3350  3376  3401  3426  3452  3477  3502  3528 

3553  3578  3603  3629  3654  3679  3704  3729  3754  3779 

3805  3830  3855  3880  3905  3930  3955  3980  4005  4030 

4055  4080  4105  4130  4155  4180  4204  4229  4254  4279 

4304  4329  4353  4378  4403  4428  4452  4477  4502  4527 

4551  4576  4601  4625  4650  4674  4699  4724  4748  4773 

4797  4822  4846  4871  4895  4920  4944  4969  4993  5018 

5042  5066  5091  5115  5139  5164  5188  5212  5237  5261 

5285  5310  5334  5358  5382  5406  6431  5455  5479  5503 

25527  25551  25575  25600  25624  25648  25672  25696  25720  25744 

5768  5792  5816  5840  5864  5888  5912  5935  5959  5983 

6007  6031  6055.  6079  6102  6126  6150  6174  6198  6221 

6245  6269  6293  6316  6340  6364  6387  6411  6435  6458 

6482  6505  6529  6553  6576  6600  6623  6647  6670  6694 

6717  6741  6764  6788  6811  6834  6858  6881  6905  6928 

6951  6975  6998  7021  7045  7068  7091  7114  7138  7161 

7184  7207  7231  7254  7277  7300  7323  7346  7370  7393 

7416  7439  7462  7485  7508  7531  7554  7577  7600  7623 

7646  7669  7692  7715  7738  7761  7784  7807  7830  7852 

27875  27898  27921  27944  27967  27989  28012  28035  28058  28081 

8103  8126  8149  8171  8194  8217  8240  8262  8285  8307 

8330  8353  8375  8398  8421  8443  8466  8488  8511  8533 

8556  8578  8601  8623  8646  8668  8691  8713  8735  8758 

8780  8803  8825  8847  8870  8892  8914  8937  8959  8981 

9003  9026  9048  9070  9092  9115  9137  9159  9181  9203 

9226  9248  9270  9292  9314  9336  9358  9380  9403  9425 

9447  9469  9491  9513  9535  9557  9579  9601  9623  9645 

9667  9688  9710  9732  9754  9776  9798  9820  9842  9863 

9885  9907  9929  9951  9973  999430016300383006030081 

30103  30125  30146  30168  30190  30211  30233  30255  30276  30298 


LOGARITHMS  OF  NUMBERS  119 


N|0123456789 


200 

1 
2 
3 

4 
5 
6 

7 


210 

1 
2 
3 
4 
5 
6 
7 
8 


30103  30125  30146  30168  30190  30211  30233  30255  30276  30298 

032&  0341  0363  0384  0406  0428  0449  0471  0492  0514 

0535  0557  0578  0600  0621  0643  0664  068*5  0707  0728 

0750  0771  0792  0814  0835  0856  0878  0899  0920  0942 

0963  0984  1006  1027  1048  1069  1091  1112  1133  1154 

1175  1197  1218  1239  1260  1281  1302  1323  1345  1366 

1387  1408  1429  1450  1471  1492  1513  1634  1555  1576 

1597  1618  1639  1660  1681  1702  1723  1744  1765  1785 

1806  1827  1848  1869  1890  1911  1931  1952  1973  1994 

2015  2035  2056  2077  2098  2118  2139  2160  2181  2201 

32222  32243  32263  32284  32305  32325  32346  32366  32387  32408 

2428  2449  2469  2490  2510  2531  2552  2572  2593  2613 

2634  2654  2675  2695  2715  2736  2756  2777  2797  2818 

2838  2858  2879  2899  2919  2940  2960  2980  3001  3021 

3041  3062  3082  3102  3122  3143  3163  3183  3203  3224 

3244  3264  3284  3304  3325  3345  3365  3385  3405  3425 

3445  3465  3486  3506  3526  3546  3566  3586  3606  3626 

3646  3666  3686  3706  3726  3746  3766  3786  3806  3826 

3846  3866  3885  3905  3925  3945  3965  3985  4005  4025. 

4044  4064  4084  4104  4124  4143  4163  4183  4203  4223 

34242  34262  34282  34301  34321  34341  34361  34380  34400  34420 

4439  4459  4479  4498  4518  4537  4557  4577  4596  4616 

4635  4655  4674  4694  4713  4733  4753  4772  4792  4811 

4830  4850  4869  4889  4908  4928  4947  4967  4986  5005 

5025  5044  5064  5083  5102  5122  5141  5160  5180  5199 

5218  5238  5257  5276  5295  5315  5334  5353  5372  5392 

5411  5430  5449  5468  5488  5507  5526  5545  5564  5583 

5603  5622  5641  5660  5679  5698  5717  5736  5755  5774 

5793  5813  5832  5851  5870  5889  5908  5927  5946  5965 

5984  6003  6021  6040  6059  6078  6097  6116  6135  6154 

36173  36192  36211  36229  36248  36267  36286  36305  36324  36342 

6361  6380  6399  6418  6436  6455  6474  6493  6511  6530 

6549  6568  6586  6605  6624  6642  6661  6680  6698  6717 

6736  6754  6773  6791  6810  6829  6847  6866  6884  6903 

6922  6940  6959  6977  6996  7014  7033  7051  7070  7088 

7107  7125  7144  7162  7181  7199  7218  7236  7254  7273 

7291  7310  7328  7346  7365  7383  7401  7420  7438  7457 

7475  7493  7511  7530  7548  7566  7585  7603  7621  7639 

7658  7676  7694  7712  7731  7749  7767  7785  7803  7822 

7840  7858  7876  7894  7912  7931  7949  7967  7985  8003 

38021  38039  38057  38075  38093  38112  38130  38148  38166  38184 

8202  8220  8238  8256  8274  8292  8310  8328  8346  8364 

8382  8399  8417  8435  8453  8471  8489  8507  8525  8543 

8561  8578  8596  8614  8632  8650  8668  8686  8703  8721 

8739  8757  8775  8792  8810  8828  8846  8863  8881  8899 

8917  8934  8952  8970  8987  9005  9023  9041  9058  9076 

9094  9111  9129  9146  9164  9182  9199  9217  9235  9252 

9270  9287  9305  9322  9340  9358  9375  9393  9410  9428 

9445  9463  9480  9498  9515  9533  9550  9568  9585  9602 

9620  9637  9655  9672  9690  9707  9724  9742  9759  9777 


120  LOGARITHMS  OF  NUMBERS 


N 

250 

1 
2 
3 
4 
5 
6 
7 


260 

1 
2 
3 

4 
5 
6 
7 
8 
9 

270 

1 
2 
3 
4 
5 
6 
7 
8 
9 

280 
1 
2 
3 
4 
5 
6 
7 


290 
1 
2 
3 
4 
6 
6 
7 
8 


300 


678 


39794  39811  39829  39846  39863  39881  39898  39915  39933  39950 

9967  9985  40002  40019  40037  40054  40071  40088  40106  40123 

4014040157  0175  0192  0209  0226  0243  0261  0278  0295 

0312  0329  0346  0364  0381  0398  0415  0432  0449  0466 

0483  0500  0518  0535  0552  0569  0586  0603  0620  0637 

0654  0671  0688  0705  0722  0739  0756  0773  0790  0807 

0824  0841  0858  0875  0892  0909  0926  0943  0960  0976 

0993  1010  1027  1044  1061  1078  1095  1111  1128  1145 

1162  1179  1196  1212  1229  1246  1263  1280  1296  1313 

1330  1347  1363  1380  1397  1414  1430  1447  1464  1481 

41497  41514  41531  41547  41564  41581  41597  41614  41631  41647 

1664  1681  1697  1714  1731  1747  1764  1780  1797  1814 

1830  1847  1863  1880  1896  1913  1929  1946  1963  1979 

1996  2012  2029  2045  2062  2078  2095  2111  2127  2144 

2160  2177  2193  2210  2226  2243  2259  2275  2292  2308 

2325  2341  2357  2374  2390  2406  2423  2439  2455  2472 

2488  2504  2521  2537  2553  2570  2586  2602  2619  2635 

2651  2667  2684  2700  2716  2732  2749  2765  2781  2797 

2813  2830  2846  2862  2878  2894  2911  2927  2943  2959 

2975  2991  3008  3024  3040  3056  3072  3088  3104  3120 

43136  43152  43169  43185  43201  43217  43233  43249  43265  43281 

3297  3313  3329  3345  3361  3377  3393  3409  3425  3441 

3457  3473  3489  3505  3521  3537  3553  3569  3584  3600 

3616  3632  3648  3664  3680  3696  3712  3727  3743  3759 

3775  3791  3807  3823  3838  3854  3870  3886  3902  3917 

3933  3949  3965  3981  3996  4012  4028  4044  4059  4075 

4091  4107  4122  4138  4154  4170  4185  4201  4217  4232 

4248  4264  4279  4295  4311  4326  4342  4358  4373  4389 

4404  4420  4436  4451  4467  4483  4498  4514  4529  4545 

4560  4576  4592  4607  4623  4638  4654  4669  4685  4700 

44716  44731  44747  44762  44778  44793  44809  44824  44840  44855 

4871  4886  4902  4917  4932  4948  4963  4979  4994  5010 

5025  5040  5056  5071  5086  5102  5117  5133  5148  5163 

5179  5194  5209  5225  5240  5255  5271  5286  5301  5317 

5332  5347  5362  5378  5393  5408  5423  5439  5454  5469 

5484  5500  5515  5530  5545  5561  5576  5591  5606  5621 

5637  5652  5667  5682  5697  5712  5728  5743  5758  5773 

5788  5803  5818  5834  5849  5864  5879  5894  5909  5924 

5939  5954  5969  5984  6000  6015  6030  6045  6060  6075 

6090  6105  6120  6135  6150  6165  6180  6195  6210  6225 

46240  46255  46270  46285  46300  46315  46330  46345  46359  46374 

6389  6404  6419  6434  6449  6464  6479  6494  6509  6523 

6538  6553  6568  6583  6598  6613 .  6627  6642  6657  6672 

6687  6702  6716  6731  6746  6761  6776  6790  6805  6820 

6835  6850  6864  6879  6894  6909  6923  6938  6953  6967 

6982  6997  7012  7026  7041  7056  7070  7085  7100  7114 

7129  7144  7159  7173  7188  7202  7217  7232  7246  7261 

7276  7290  7305  7319  7334  7349  7363  7378  7392  7407 

7422  7436  7451  7465  7480  7494  7509  7524  7538  7553 

7567  7582  7596  7611  7625  7640  7654  7669  7683  7698 

47712  47727  47741  47756  47770  47784  47799  47813  47828  47842 


LOGARITHMS  OF  NUMBERS  121 


6789 


300  47712  47727  47741  47756  47770  47784  47799  47813  47828  47842 


1 


310 

1 
2 
3 

4 
5 
6 

7 
8 


320 
1 
2 
3 

4 
5 
6 
7 
8 
9 

330 

1 
2 
3 
4 
5 
6 
7 
8 
9 

340 

1 

2 
3 
4 
5 
6 
7 


350 


7857  7871  7885  7900  7914  7929  7943  7958  7972  7986 


2  8001  8015  8029  8044  8058  8073  8087  8101  8116  8130 

3  I  8144  8159  8173  8187  8202  8216  8230  8244  8259  8273 


8287  8302  8316  8330  8344  8359  8373  8387  8401  8416 

8430  8444  8458  8473  8487  8501  8515  8530  8544  8558 

8572  8586  8601  8615  8629  8643  8657  8671  8686  8700 

8714  8728  8742  8756  8770  8785  8799  8813  8827  8841 

8855  8869  8883  8897  8911  8926  8940  8954  8968  8982 

8996  9010  9024  9038  9052  9066  9080  9094  9108  9122 

49136  49150  49164  49178  49192  49206  49220  49234  49248  49262 

9276  9290  9304  9318  9332  9346  9360  9374  9388  9402 

9415  9429  9443  9457  9471  9485  9499  9513  9527  9541 

9554  9568  9582  9596  9610  9624  9638  9651  9665  9679 

9693  9707  9721  9734  9748  9762  9776  9790  9803  9817 

9831  9845  9859  9872  9886  9900  9914  9927  9941  9955 

9969  9982  9996  50010  50024  50037  50051  50065  50079  50092 

50106  50120  50133  0147  0161  0174  0188  0202  0215  0229 

0243  0256  0270  0284  0297  0311  0325  0338  0352  0365 

0379  0393  0406  0420  0433  0447  0461  0474  0488  0501 

50515  50529  50542  50556  60569  50583  50596  50610  50623  50637 

0651  0664  0678  0691  0705  0718  0732  0745  0759  0772 

0786  0799  0813  0826  0840  0853  0866  0880  0893  0907 

0920  0934  0947  0961  0974  0987  1001  1014  1028  1041 

1055  1068  1081  1095  1108  1121  1135  1148  1162  1175 

1188  1202  1215  1228  1242  1255  1268  1282  1295  1308 

1322  1335  1348  1362  1375  1388  1402  1415  1428  1441 

1455  1468  1481  1495  1508  1521  1534  1548  1561  1574 

1587  1601  1614  1627  1640  1654  1667  1680  1693  1706 

1720  1733  1746  1759  1772  1786  1799  1812  1825  1838 

51851  51865  51878  51891  51904  51917  51930  51943  51957  51970 

1983  1996  2009  2022  2035  2048  2061  2075  2088  2101 

2114  2127  2140  2153  2166  2179  2192  2205  2218  2231 

2244  2257  2270  2284  2297  2310  2323  2336  2349  2362 

2375  2388  2401  2414  2427  2440  2453  2466  2479  2492 

2504  2517  2530  2543  2556  2569  2582  2595  2608  2621 

2634  2647  2660  2673  2686  2699  2711  2724  2737  2750 

2763  2776  2789  2802  2815  2827  2840  2853  2866  2879 

2892  2905  2917  2930  2943  2956  2969  2982  2994  3007 

3020  3033  3046  3058  3071  3084  3097  3110  3122  3135 

53148  53161  53173  53186  53199  53212  53224  53237  53250  53263 

3275  3288  3301  3314  3326  3339  3352  3364  3377  3390 

3403  3415  3428  3441  3453  3466  3479  3491  3504  3517 

3529  3542  3555  3567  3580  3593  3605  3618  3631  3643 

3656  3668  3681  3694  3706  3719  3732  3744  3757  3769 

3782  3794  3807  3820  3832  3845  3857  3870  3882  3895 

3908  3920  3933  3945  3958  3970  3983  3995  4008  4020 

4033  4045  4058  4070  4083  4095  4108  4120  4133  4145 

4158  4170  4183  4195  4208  4220  4233  4245  4258  4270 

4283  4295  4307  4320  4332  4345  4357  4370  4382  4394 

54407  54419  54432  54444  54456  54469  54481  54494  54506  54518 


122  LOGARITHMS  OF  NUMBERS. 


NO12345678 


350 

1 
2 
3 
4 
5 
6 
7 
8 
9 

360 

1 
2 
3 
4 
5 
6 
7 
8 


370 

1 
2 

3 
4 
5 
6 

7 
8 
9 

380 

1 
2 
3 
4 
5 
6 
7 
8 
9 


3 
4 
5 
6 
7 
8 
9 

400 


54407  64419  54432  54444  54456  54469  54481  54494  54506  54518 

4531  4543  4555  4568  4580  4593  4605  4617  4630  4642 

4654  4667  4679  4691  4704  4716  4728  4741  4753  4765 

4777  4790  4802  4814  4827  4839  4851  4864  4876  4888 

4900  4913  4925  4937  4949  4962  4974  4986  4998  5011 

5023  5035  5047  5060  5072  5084  5096  5108  5121  5133 

5145  5157  5169  5182  5194  5206  5218  5230  5242  5255 

5267  5279  5291  5303  5315  5328  5340  5352  5364  5376 

5388  6400  5413  5425  5437  5449  5461  5473  5485  6497 

5509  5522  5534  5546  5558  6570  5582  5594  5606  5618 

55630  55642  55654  55666  55678  55691  55703  55715  55727  55739 

5751  5763  5775  6787  5799  6811  6823  5835  5847  5859 

5871  5883  5895  5907  5919  5931  6943  5955  5967  5979 

5991  6003  6015  6027  6038  6050  6062  6074  6086  6098 

6110  6122  6134  6146  6158  6170  6182  6194  6205  6217 

6229  6241  6253  6265  6277  6289  6301  6312  6324  6336 

6348  6360  6372  6384  6396  6407  6419  6431  6443  6455 

6467  6478  6490  6502  6514  6526  6538  6549  6561  6573 

6585  6597  6608  6620  6632  6644  6656  6667  6679  6691 

6703  6714  6726  6738  6750  6761  6773  6785  6797  6808 

56820  56832  56844  56855  56867  56879  56891  56902  56914  56926 

6937  6949  6961  6972  6984  6996  7008  7019  7031  7043 

7054  7066  7078  7089  7101  7113  7124  7136  7148  7159 

7171  7183  7194  7206  7217  7229  7241  7252  7264  7276 

7287  7299  7310  7322  7334  7345  7357  7368  7380  7392 

7403  7415  7426  7438  7449  7461  7473  7484  7496  7507 

7519  7530  7542  7553  7565  7576  7588  7600  7611  7623 

7634  7646  7657  7669  7680  7692  7703  7715  7726  7738 

7749  7761  7772  7784  7795  7807  7818  7830  7841  7852 

7864  7875  7887  7898  7910  7921  7933  7944  7955  7967 

57978  57990  58001  58013  58024  58035  58047  58058  58070  58081 

8092  8104  8115  8127  8138  8149  8161  8172  8184  8195 

8206  8218  8229  8240  8252  8263  8274  8286  8297  8309 

8320  8331  8343  8354  8365  8377  8388  8399  8410  8422 

8433  8444  8456  8467  8478  8490  8501  8512  8524  8535 

8546  8557  8569  8580  8591  8602  8614  8625  8636  8647 

8659  8670  8681  8692  8704  8715  8726  8737  8749  8760 

8771  8782  8794  8805  8816  8827  8838  8850  8861  8872 

8883  8894  8906  8917  8928  8939  8950  8961  8973  8984 

8995  9006  9017  9028  9040  9051  9062  9073  9084  9095 

59106  59118  59129  59140  59151  59162  59173  59184  59195  59207 

9218  9229  9240  9251  9262  9273  9284  9295  9306  9318 

9329  9340  9351  9362  9373  9384  9395  9406  9417  9428 

9439  9450  9461  9472  9483  9494  9506  9517  9528  9539 

9550  9561  9572  9583  9594  9605  9616  9627  9638  9649 

9660  9671  9682  9693  9704  9715  9726  9737  9748  9759 

9770  9780  9791  9802  9813  9824  9835  9846  9857  9868 

9879  9890  9901  9912  9923  9934  9945  9956  9966  9977 

9988  9999  60010  60021  60032  60043  60054  60065  60076  60086 

60097  60108  0119  0130  0141  0152  0163  0173  0184  0195 

60206  60217  60228  60239  60249  60260  60271  60282  60293  60304 


LOGARITHMS  OF  NUMBERS  123 


NO12345678 


400 
1 

2 
3 

4 
5 


9 

410 

1 
2 
3 

4 
5 
6 
7 
8 
9 

420 

1 

2 
3 
4 
5 
6 
7 
8 
9 

430 
1 

2 
3 
4 
5 
6 
7 
8 
9 

440 

1 

2 
3 

4 
5 
6 

7 
8 
9 

450 


60206  60217  60228  60239  60249  60260  60271  60282  60293  60304 

0314  0325  0336  0347  0358  0369  0379  0390  0401  0412 

0423  0433  0444  0455  0466  0477  0487  0498  0509  0520 

0531  0541  0552  0563  0574  0584  0595  0606  0617  0627 

0638  0649  0660  0670  0681  0692  0703  0713  0724  0735 

0746  0756  0767  0778  0788  0799  0810  0821  0831  0842 

0853  0863  0874  0885  0895  0906  0917  0927  0938  0949 

0969  0970  0981  0991  1002  1013  1023  1034  1045  1055 

1066  1077  1087  1098  1109  1119  1130  1140  1151  1162 

1172  1183  1194  1204  121^  1225  1236  1247  1257  1268 

61278  61289  61300  61310  61321  61331  61342  61352  61363  61374 

1384  1395  1405  1416  1426  1437  1448  1458  1469  1479 

1490  1500  1511  1521  1532  1542  1553  1563  1574  1584 

1595  1606  1616  1627  1637  1648  1658  1669  1679  1690 

1700  1711  1721  1731  1742  1752  1763  1773  1784  1794 

1805  1815  1826  1836  1847  1857  1868  1878  1888  1899 

1909  1920  1930  1941  1951  1962  1972  1982  1993  2003 

2014  2024  2034  2045 '  2055  2066  2076  2086  2097  2107 

2118  2128  2138  2149  2159  2170  2180  2190  2201  2211 

2221  2232  2242  2252  2263  2273  2284  2294  2304  2315 

62325  62335  62346  62356  62366  62377  62387  62397  62408  62418 

2428  2439  2449  2459  2469  2480  2490  2500  2511  2521 

2531  2542  2552  2562  2572  2583  2593  2603  2613  2624 

2634  2644  2655  2665  2675  2685  2696  2706  2716  2726 

2737  2747  2757  2767  2778  2788  2798  2808  2818  2829 

2839  2849  2859  2870  2880  2890  2900  2910  2921  2931 

2941  2951  2961  2972  2982  2992  3002  3012  3022  3033 

3043  3053  3063  3073  3083  3094  3104  3114  3124  3134 

3144  3155  3165  3175  3185  3195  3205  3215  3225  3236 

3246  3256  3266  3276  3286  3296  3306  3317  3327  3337 

63347  63357  63367  63377  63387  63397  63407  63417  63428  63438 

3448  3458  3468  3478  3488  3498  3508  3518  3528  3538 

3548  3558  3568  3579  3589  3599  3609  3619  3629  3639 

3649  3659  3669  3679  3689  3699  3709  3719  3729  3739 

3749  3759  3769  3779  3789  3799  3809  3819  3829  3839 

3849  3859  3869  3879  3889  3899  3909  3919  3929  3939 

3949  3959  3969  3979  3988  3998  4008  4018  4028  4038 

4048  4058  4068  4078  4088  4098  4108  4118  4128  4137 

4147  4157  4167  4177  4187  4197  4207  4217  4227  4237 

4246  4256  4266  4276  4286  4296  4306  4316  4326  4335 

64345  64355  64365  64375  64385  64395  64404  64414  64424  64434 

4444  4454  4464  4473  4483  4493  4503  4513  4523  4532 

4542  4552  4562  4572  4582  4591  4601  4611  4621  4631 

4640  4650  4660  4670  4680  4689  4699  4709  4719  4729 

4738  4748  4758  4768  4777  4787  4797  4807  4816  4826 

4836  4846  4856  4865  4875  4885  4895  4904  4914  4924 

4933  4943  4953  4963  4972  4982  4902  5002  5011  6021 

5031  5040  5050  5060  5070  6079  5089  5099  5108  5118 

5128  6137  5147  6157  5167  6176  5186  5196  5205  6215 

5225  6234  5244  6254  6263  6273  5283  5292  5302  6312 

65321  65331  65341  65350  65360  65369  65379  65389  65398  65408 


124  LOGARITHMS  OF  NUMBERS 

1 


450 

1 
2 
3 

4 
5 
6 

7 

3 

a 

460 
1 
2 
3 

4 
5 
6 

7 
9 


470 
1 
2 
3 

4 
5 
6 
7 
8 
9 

480 

1 
2 
3 

4 
5 
6 
7 
8 
9 

490 
1 
2 
3 
4 
5 
6 
7 
8 


500 


3   4:   5   6   7   8   9 


65321  65331  65341  65350  65360  65369  65379  65389  65398  65408 

5418  5427  5437  5447  5456  5466  5475  5485  5495  5504 

5514  5523  5533  5543  5552  5562  5571  5581  5591  5600 

5610  5619  5629  5639  5648  5658  5667  5677  5686  5696 

5706  5715  5725  5734  5744  5753  5763  6772  5782  5792 

5801  5811  5820  5830  5839  5849  5858  5868  6877  5887 

5896  5906  5916  5925  5935  5944  5954  5963  5973  5982 

5992  6001  6011  6020  6030  6039  6049  6058  6068  6077 

6087  6096  6106  6115  6124  6134  6143  6153  6162  6172 

6181  6191  6200  6210  6219  6229  6238  6247  6257  6266 

66276  66285  66295  66304  66314  66323  66332  66342  66351  66361 

6370  6380  6389  6398  6408  6417  6427  6436  6445  6455. 

6464  6474  6483  6492  6502  6511  6521  6530  6539  6549 

6558  6567  6577  6586  6596  6605  6614  6624  6633  6642 

6652  6661  6671  6680  6689  6699  6708  6717  6727  6736 

6745  6755  6764  6773  6783  6792  6801  6811  6820  6829 

6839  6848  6857  6867  6876  6885  6894  6904  6913  6922 

6932  6941  6950  6960  6969  6978  6987  6997  7006  7015 

7025  7034  7043  7052  7062  7071  7080  7089  7099  7108 

7117  7127  7136  7145  7154  7164  7173  7182  7191  7201 

67210  67219  67228  67237  67247  67256  67265  67274  67284  67293 

7302  7311  7321  733"0  7339  7348  7357  7367  7376  7385 

7394  7403  7413  7422  7431  7440  7449  7459  7468  7477 

7486  7495  7504  7514  7523  7532  7541  7550  7560  7569 

7578  7587  7596  7605  7614  7624  7633  7642  7651  7660 

7669  7679  7688  7697  7706  7715  7724  7733  7742  7752 

7761  7770  7779  7788  7797  7806  7815  7825  7834  7843 

7852  7861  7870  7879  7888  7897  7906  7916  7925  7934 

7943  7952  7961  7970  7979  7988  7997  8006  8015  8024 

8034  8043  8052  8061  8070  8079  8088  8097  8106  8115 

68124  68133  68142  68151  68160  68169  68178  68187  68196  68205 

8215  8224  8233  8242  8251  8260  8269  8278  8287  8296 

8305  8314  8323  8332  8341  8350  8359  8368  8377  8386 

8395  8404  8413  8422  8431  8440  8449  8458  8467  8476 

8485  8494  8502  851.1  8520  8529  8538  8547  8556  8565 

8574  8583  8592  8601  8610  8619  8628  8637  8646  8655 

8664  8673  8681  8690  8699  8708  8717  8726  8735  8744 

8753  8762  8771  8780  8789  8797  8806  8815  8824  8833 

8842  8851  8860  8869  8878  8886  8895  8904  8913  8922 

8931  8940  8949  8958  8966  8975  8984  8993  9002  9011 

69020  69028  69037  69046  69055  69064  69073  69082  69090  69099 

9108  9117  9126  9135  9144  9152  9161  9170  9179  9188 

9197  9205  9214  9223  9232  9241  9249  9258  9267  9276 

9285  9294  9302  9311  9320  9329  9338  9346  9355  9364 

9373  9381  9390  9399  9408  9417  9425  9434  9443  9452 

9461  9469  9478  9487  9496  9504  9513  9522  9531  9539 

9548  9557  9566  9574  9583  9592  9601  9609  9618  9627 

9636  9644  9653  9662  9671  9679  9688  9697  9705  9714 

9723  9732  9740  9749  9758  9767  •  9775  9784  9793  9801 

9810  9819  9827  9836  9845  9854  9862  9871  9880  9888 

69897  69906  69914  69923  69932  69940  69949  69958  69966  69975 


LOGARITHMS  OF  NUMBERS  125 


3456789 


500 

1 
2 
3 

4 
5 
6 
7 
8 
9 

510 

1 

2 
3 

4. 
5 
6 
7 
8 
9 

520 

1 
2 
3 
4 
5 
6 
7 
8 
9 

530 

1 
2 
3 
4 
5 
6 
7 


540 

1 
2 
3 
4 
5 
6 
7 


550 


69897  69906  69914  69923  69932  69940  69949  69958  69966  69973 

9984  9992  70001  70010  70018  70027  70036  70044  70053  70062 

70070  70079  0088  0096  0105  0114  0122  0131  0140  0148 

0157  0165  0174  0183  0191  0200  0209  0217  0226  0234 

0243  0252  0260  0269  0278  0286  0295  0303  0312  0321 

0329  0338  0346  0355  0364  0372  0381  0389  0398  0406 

0415  0424  0432  0441  0449  0458  0467  0475  0484  0492 

0501  0509  0518  0526  0535  0544  0552  0561  0569  0578 

0586  0595  0603  0612  0621  0629  0638  0646  0655  0663 

0672  0680  0689  0697  0706  0714  .0723  0731  0740  0749 

70757  70766  70774  70783  70791  70800  70808  70817  70825  70834 

0842  0851  0859  0868  0876  0885  0893  0902  0910  0919 

0927  0935  0944  0952  0961  0969  0978  0986  0995  1003 

1012  1020  1029  1037  1046  1054  1063  1071  1079  1088 

1096  1105  1113  1122  1130  1139  1147  1155  1164  1172 

1181  1189  1198  1206  1214  1223  1231  1240  1248  1257 

1265  1273  1282  1290  •  1299  1307  1315  1324  1332  1341 

1349  1357  1366  1374  1383  1391  1399  1408  1416  1425 

1433  1441  1450  1458  1466  1475  1483  1492  1500  1508 

1517  1525  1533  1542  1550  1559  1567  1575  1584  1592 

71600  71609  71617  71625  71634  71642  71650  71659  71667  71675 

1684  1692  1700  1709  1717  1725  1734  1742  1750  1759 

1767  1775  1784  1792  1800  1809  1817  1825  1834  1842 

1850  1858  1867  1875  1883  1892  1900  1908  1917  1925 

1933  1941  1950  1958  1966  1975  1983  1991  1999  2008 

2016  2024  2032  2041  2049  2057  2066  2074  2082  2090 

2099  2107  2115  2123  2132  2140  2148  2156  2165  2173 

2181  2189  2198  2206  2214  2222  2230  2239  2247  2255 

22(53  2272  2280  2288  2296  2304  2313  2321  2329  2337 

2346  2354  2362  2370  2378  2387  2395  2403  2411  2419 

72428  72436  72444  72452  72460  72469  72477  72485  72493  72501 

2609  2518  2526  2534  2542  2550  2558  2567  2575  2583 

2591  2599  2607  2616  2624  2632  2640  2648  2656  2665. 

2673  2681  2689  2697  2705  2713  2722  2730  2738  2746 

2754  2762  2770  2779  '2787  2795  2803  2811  2819  2827 

2835  2843  2852  2860  2868  2876  2884  2892  2900  2908 

2916  2925  2933  2941  2949  2957  2965  2973  2981  2989 

2997  3006  3014  3022  3030  3038  3046  3054  3062  3070 

3078  3086  3094  3102  3111  3119  3127  3135  3143  3151 

3159  3167  3175  3183  3191  3199  3207  3215  3223  3231 

73239  73247  73255  73263  73272  73280  73288  732*96  73304  73312 

3320  3328  3336  3344  3352  3360  3368  3376  3384  3392 

3400  3408  3416  3424  3432  3440  3448  3456  3464  3472 

3480  3488  3496  3504  3512  3520  3528  3536  3544  3552 

3560  3568  3576  3584  3592  3600  3608  3616  3624  3632 

3640  3648  3(556  3664  3672  3679  3687  3695  3703  3711 

3719  3727  3735  3743  3751  3759  3767  3775  3783  3791 

3799  3807  3815  3823  3830  3838  3846  3854  3862  3870 

3878  3886  3894  3902  3910  3918  3926  3933  3941  3949 

3957  3965  3973  3981  3989  3997  4005.  4013  4020  4028 

74036  74044  74052  74060  74068  74076  74084  74092  74099  74107 


126 


LOGARITHMS  OF  NUMBERS 


N 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9  ! 

550 

74036 

74044 

74052 

74060 

74068  74076 

74084 

74092 

74099 

74107; 

1 

4115 

4123 

4131 

4139 

4147 

4155 

4162 

4170 

4178 

4186 

2 

4194 

4202 

4210 

4218 

4225 

4233 

4241 

4249 

4257 

4265 

3 

4273 

4280 

4288 

4296 

4304 

4312 

4320 

4327 

4335 

4343 

4 

4351 

4359 

4367 

4374 

4382 

4390 

4398 

4406 

4414 

4421 

5 

4429 

4437 

4445 

4453 

4461 

4468 

4476 

4484 

4492 

4500 

6 

4507 

4515 

4523 

4531 

4539 

4547 

4554 

4562 

4570 

4578 

7 

4586 

4593 

4601 

4609 

4617 

4624 

4632 

4640 

4648 

4656 

8 

4663 

4671 

4679 

4687 

4095 

4702 

4710 

4718 

4726 

4733 

9 

4741 

4749 

4757 

4764 

4772 

4780 

4788 

4796 

4803 

4811 

560 

74819 

74827 

74834 

74842 

74850 

74858  74865 

74873 

74881 

74889 

1 

4896 

4904 

4912 

4920 

4927 

4935 

4943 

4950 

4958 

4966 

2 

4974 

4981 

4989 

4997 

5005 

5012 

5020 

5028 

5035 

5043 

3 

5051 

5059 

5066 

5074 

5082 

5089 

5097 

5105 

5113 

5120 

4 

5128 

5136 

5143 

5151 

5159 

5166 

5174 

5182 

5189 

5197 

5 

5205 

5213 

5220 

5228 

5236 

5243 

5251 

5259 

5266 

5274 

6 

5282 

5289 

5297 

5305 

5312 

5320 

5328 

5335 

5343 

5351 

7 

5358 

5366 

5374 

5381 

5389 

5397 

5404 

5412 

5420 

5427 

8 

5435 

5442 

5450 

5458 

5465 

5473 

5481 

5488 

5496 

5504 

9 

5511 

5519 

5526 

5534 

5542 

5549 

5557 

5565 

5572 

5580 

670 

75587 

75595 

75603 

75610 

75618 

75626 

75633 

75641 

75648 

75656 

1 

5664 

5671 

5679 

5686 

5694 

5702 

5709 

5717 

5724 

5732 

2 

5740 

5747 

5755 

5762 

5770 

5778 

5785 

5793 

5800 

5808 

3 

5815 

5823 

5831 

5838 

5846 

5853 

5861 

5868 

5876 

5884 

4 

5891 

5899 

5906 

5914 

5921 

5929 

5937 

5944 

5952 

5959 

5 

5967 

5974 

6982 

5989 

5997 

6005 

6012 

6020 

6027 

6035 

6 

6042 

6050 

6057 

6065 

6072 

6080 

6087 

6095 

6103 

6110 

7 

6118 

6125 

6133 

6140 

6148 

6155 

6163 

6170 

6178 

6185 

8 

6193 

6200 

6208 

6215 

6223 

6230 

6238 

6245 

6253 

6260 

9 

6268 

6275 

6283 

6290 

6298 

6305 

6313 

6320 

6328 

6335 

580 

76343 

76350 

76358 

76365 

76373  76380  76388 

76395 

76403 

76410 

1 

6418 

6425 

6433 

6440 

6448 

6455 

6462 

6470 

6477 

6485 

2 

6492 

6500 

6507 

6515 

6522 

6530 

6537 

6545 

6552 

6559 

3 

6567 

6574 

6582 

6589 

6597 

6604 

6612 

6619 

6626 

6634  ' 

4 

6641 

6649 

6656 

6664 

6671 

6678 

6686 

6693 

6701 

6708 

5 

6716 

6723 

6730 

6738 

6745 

6753 

6760 

6768 

6775 

6782 

6 

6790 

6797 

6805 

6812 

6819 

6827 

6884 

6842 

6849 

6856 

7 

6864 

6871 

6879 

6886 

6893 

6901 

6908 

6916 

6923 

6930 

8 

6938 

6945 

6953 

6960 

6967 

6975 

6982 

6989 

6997 

7004 

9 

7012 

7019 

7026 

7034 

7041 

7048 

7056 

7063 

7070 

7078 

590 

77085 

77093 

77100 

77107 

77115 

77122 

77129 

77137 

77144 

77151 

1 

7159 

7166 

7173 

7181 

7188 

7195 

7203 

7210 

7217 

7225 

2 

7232 

7240 

7247 

7254 

7262 

7269 

7276 

7283 

7291 

7298 

3 

7305 

7313 

7320 

7327 

7335 

7342 

7349 

7357 

7364 

7371 

4 

7379 

7386 

7393 

7401 

7408 

7415 

7422 

7430 

7437 

7444 

5 

7452 

7459 

7466 

7474 

7481 

7488 

7495 

7503 

7510 

7517 

6 

7525 

7532 

7539 

7546 

7554 

7561 

7568 

7576 

7583 

7590 

7 

7597 

7605 

7612 

7619 

7627 

7634 

7641 

7648 

7656 

7663 

8 

7670 

7677 

7685 

7692 

7699 

7706 

7714 

7721 

7728 

7735 

9 

7743 

7750 

7757 

7764 

7772 

7779 

7786 

7793 

7801 

7808 

600 

77815 

77822 

77830 

77837 

77844 

77851 

77859 

77866 

77873 

77880 

LOGARITHMS  OF  NUMBERS  127 


N 

600 
1 

2 
3 

4 
5 
6 
7 
8 
9 

610 
1 


3 


6 

7 
8 
9 

620 
1 

2 
3 
4 
5 
6 
7 
8 
9 

680 
1 
2 
3 
4 
5 
6 
7 
8 
9 

640 
1 
2 
3 

4 
5 


.  7 


650 


6789 


77815  77822  77830  77837  77844  77851  77859  77866  77873  77880 

7887  7895  7902  7909  7916  7924  7931  7938  7945  7952 

7960  7967  7974  7981  7988  7996  8003  8010  8017  8025 

8032  8039  8046  8053  8061  8068  8075  8082  8089  8097 

8104  8111  8118  8125  8132  8140  8147  8154  8161  8168 

8176  8183  8190  8197  8204  8211  8219  8226  8233  8240 

8247  8254  8262  8269  8276  8283  8290  8297  8305.  8312 

8319  8326  8333  8340  8347  8355  8362  8369  8376  8383 

8390  8398  8405  8412  8419  8426  8433  8440  8447  8455. 

8462  8469  8476  8483  8490  8497  8504  8512  8519  8526 

78533  78540  78547  78554  78561  78569  78576  78583  78590  78597 

8604  8611  8618  8625  8633  8640  8647  8654  8661  8668 

8675  8682  8689  8696  8704  8711  8718  8725  8732  8739 

8746  8753  8760  8767  8774  8781  8789  8796  8803  8810 


4  I  8817  8824  8831  8838  8845  8852  8859  8866  8873  8880 

5  8888  8895  8902  8909  8916  8923  8930  8937  8944  8951 


8958  8965  8972  8979  8986  8993  9000  9007  9014  9021 

9029  9036  9043  9050  9057  9064  9071  9078  9085  9092 

9099  9106  9113  9120  9127  9134  9141  9148  9155  9162 

9169  9176  9183  9190  9197  9204  9211  9218  9225  9232 

79239  79246  79253  79260  79267  79274  79281  79288  79295  79302 

9309  9316  9323  9330  9337  9344  9351  9358  9365  9372 

9379  9386  9393  9400  9407  9414  9421  9428  9435  9442 

9449  9456  9463  9470  9477  9484  9491  9498  9505  9511 

9518  9525  9532  9539  9546  9553  9560  9567  9574  9581 

9588  9595  9602  9609  9616  9623  9630  9637  9644  9650 

9657  9664  9671  9678  9685  9692  9699  9706  9713  9720 

9727  9734  9741  9748  9754  9761  9768  9775  9782  9789 

9796  9803  9810  9817  9824  9831  9837  9844  9851  9858 

9865  9872  9879  9886  9893  9900  9906  9913  9920  9927 

79934  79941  79948  79955  79962  79969  79975  79982  79989  79996 

80003  80010  80017  80024  80030  80037  80044  80051  80058  80065 

0072  0079  0085  0092  0099  0106  0113  0120  0127  0134 

0140  0147  0154  0161  0168  0175  0182  0188  0195  0202 

0209  0216  0223  0229  0236  0243  0250  0257  0264  0271 

0277  0284  0291  0298  0305  0312  0318  0325  0332  0339 

0346  0353  0359  0366  0373  0380  0387  0393  0400  0407 

0414  0421  0428  0434  0441  0448  0455  0462  0468  0475 

0482  0489  0496  0502  0509  0516  0523  0530  0536  0543 

0550  0557  0564  0570  0577  0584  0591  0598  0604  0611 

80618  80625  80632  8063P  80645  80652  80659  80665  80672  80679 

0686  0693  0699  0706  0713  0720  0726  0733  0740  0747 

0754  0760  0767  0774  0781  0787  0794  0801  0808  0814 

0821  0828  0835  0841  0848  0855  0862  0868  0875  0882 

0889  0895  0902  0909  0916  0922  0929  0936  0943  0949 

0956  0963  0969  0976  0983  0990  0996  1003  1010  1017 

1023  1030  1037  1043  1050  1057  1064  1070  1077  1084 

1090  1097  1104  1111  1117  1124  1131  1137  1144  1151 

1158  1164  1171  1178  1184  1191  1198  1204  1211  1218 

1224  1231  1238  1245  1251  1258  1265  1271  1278  1285 

81291  81298  81305  81311  81318  81325  81331  81338  81345  81351 


128  LOGARITHMS  OF  NUMBERS 


0123456789 


650 

1 
2 
3 
4 
5 


680 
1 
2 
3 
4 
5 
6 
7 
8 
9 

670 
1 
2 
3 

4 
5 
6 
7 
8 
9 

680 
1 
2 
3 
4 
5 
6 
7 
8 
9 

690 


81291  81298  81305  81311  81318  81325  81331  81338  81345  81351 

1358  1365  1371  1378  1385  1391  1398  1405  1411  1418 

1425  1431  1438  1445  1451  1458  1465  1471  1478  1485 

1491  1498  1505  1511  1518  1525  1531  1538  1544  1551 

1558  1564  1571  1578  1584  1591  1598  1604  1611  1617 

1624  1631  1637  1644  1651  1657  1664  1671  1677  1684 

1690  1697  1704  1710  1717  1723  1730  1737  1743  1750 

1757  1763  1770  1776  1783  1790  1796  1803  1809  1816 

1823  1829  1836  1842  1849  1856  1862  1869  1875  1882 

1889  1895  1902  1908  1915  1921  1928  1935  1941  1948 

81954  81961  81968  81974  81981  81987  81994  82000  82007  82014 

2020  2027  2033  2040  2046  2053  2060  2066  2073  2079 

2086  2092  2099  2105  2112  2119  2125  2132  2138  2145 

2151  2158  2164  2171  2178  2184  2191  2197  2204  2210 

2217  2223  2230  2236  2243  2249  2256  2263  2269  2276 

2282  2289  2295  2302  2308  2315  2321  2328  2334  2341 

2347  2354  2360  2367  2373  2380  2387  2393  2400  2406 

2413  2419  2426  2432  2439  2445  2452  2458  2465  2471 

2478  2484  2491  2497  2504  2510  2517  2523  2530  2536 

2543  2549  2556  2562  2569  2575  2582  2588  2595  2601 

82607  82614  82620  82627  82633  82640  82646  82653  82659  82666 

2672  2679  2685  2692  2698  2705  2711  2718  2724  2730 

2737  2743  2750  2756  2763  2769  2776  2782  2789  2795 

2802  2808  2814  2821  2827  2834  2840  2847  2853  2860 

2866  2872  2879  2885  2892  2898  2905  2911  2918  2924 

2930  2937  2943  2950  2956  2963  2969  2975  2982  2988 

2995  3001  3008  3014  3020  3027  3033  3040  3046  3052 

3059  3065  3072  3078  3085  3091  3097  3104  3110  3117 

3123  3129  3136  3142  3149  3155  3161  3168  3174  3181 

3187  3193  3200  3206  3213  3219  3225  3232  3238  3245 

83251  83257  83264  83270  83276  83283  83289  83296  83302  83308 

3315  3321  3327  3334  3340  3347  3353  3359  3366  3372 

3378  3385  3391  3398  3404  3410  3417  3423  3429  3436 

3442  3448  3455  3461  3467  3474  3480  3487  3493  3499 

3506  3512  3518  3525  3531  3537  3544  3550  3556  3563 

3569  3575  3582  3588  3594  3601  3607  3613  3620  3626 

3632  3639  3645  3651  3658  3664  3670  3677  3683 

3696  3702  3708  3715  3721  3727  3734  3740  3746  3753 

3759  3765  3771  3778  3784  3790  3797  3803  3809  3816 

3822  3828  3835  3841  3847  3853  3860  3866  3872  3879 

83885  83891  83897  83904  83910  83916  83923  83929  83935  83942 

3948  3954  3960  3967  3973  3979  3985  3992  3998  4004 

4011  4017  4023  4029  4036  4042  4048  4055  4061  4067 

4073  4080  4086  4092  4098  4105  4111  4117  4123  4130 

4136  4142  4148  4155  4161  4167  4173  4180  4186  4192 

4198  4205  4211  4217  4223  4230  4236  4242  4248  4255 

4261  4267  4273  4280  4286  4292  4298  4305  4311  4317 

4323  4330  4336  4342  4348  4354  4361  4367  4373  4379 

4386  4392  4398  4404  4410  4417  4423  4429  4435  4442 

4448  4454  4460  4466  4473  4479  4485  4491  4497  4504 


700  84510  84516  84522  84528  84535  84541  84547  84553  84559  8456(3 


LOGARITHMS  OF  NUMBERS  129 


N 


3456789 


700 

1 
2 
3 
4 
5 
6 
7 
8 
9 

710 

1 
2 
3 

4 
5 
6 
7 
8 
9 

720 

1 
2 

3 
4 
5 
6 

7 


730 

1 
2 
3 
4 
5 
6 
7 
8 
9 

740 

1 
2 
3 
4 
5 
6 
7 
8 
9 

750 


84510  84516  84522  84528  84535  84541  84547  84553  84559  84566 
4572  4578  4584  4590  4597  4603  4609  4615  4621  4628 

4634  -4640  4646  4652  4658  4665  4671  4677  4683  4689 

4696  4702  4708  4714  4720  4726  4733  4739  4745  4751 

4757  4763  4770  4776  4782  4788  4794  4800  4807  4813 

4819  4825  4831  4837  4844  4850  4856  4862  4868  4874 

4880  4887  4893  4899  4905  4911  4917  4924  4930  4936 

4942  4948  4954  4960  4967  4973  4979  4985  4991  4997 

5003  5009  5016  5022  5028  5034  5040  5046  5052  5058 

5065  5071  5077  5083  5089  5095  5101  5107  5114  5120 

85126  85132  85138  85144  85150  85156  85163  85169  85175  85181 

5187  5193  5199  5205  5211  5217  5224  5230  5236  5242 

5248  5254  5260  5266  5272  5278  5285  5291  5297  5303 

5309  5315  5321  5327  5333  5339  5345  5352  5358  5364 

5370  5376  5382  5388  5394  5400  5406  5412  5418  5425. 

5431  5437  5443  5449  5455  5461  *467  5473  5479  5485 

5491  5497  5503  5509  '5516  5522  6528  5534  5540  5546 

5552  5558  5564  5570  5576  5582  5588  5594  5600  5606 

5612  5618  5625  5631  5637  5643  5649  5655  5661  5667 

5673  5679  5685  5691  5697  5703  5709  5715  5721  5727 

85733  85739  85745  85751  85757  85763  85769  85775  85781  85788 

5794  5800  5806  5812  5818  5824  5830  5836  5842  5848 

5854  5860  5866  5872  5878  5884  5890  5896  5902  6908 

5914  5920  5926  5932  5938  5944  5950  5956  5962  5968 

5974  5980  5986  5992  5998  6004  6010  6016  6022  6028 

6034  6040  6046  6052  6058  6064  6070  6076  6082  6088 

6094  6100  6106  6112  6118  6124  6130  6136  6141  6147 

6153  6159  6165  6171  6177  6183  6189  6195  6201  6207 

6213  6219  6225  6231  6237  6243  6249  6255  6261  6267 

6273  6279  6285  6291  6297  6303  6308  6314  6320  6326 

86332  86338  86344  86350  86356  86362  86368  86374  86380  86386 

6392  6398  6404  6410  6415  6421  6427  6433  6439  6445 

6451  6457  6463  6469  6475  6481  6487  6493  6499  6504 

6510  6516  6522  6528  6534  6540  6546  6552  6558  6564 

6570  6576  6581  6587  6593  6599  6605  6611  6617  6623 

6629  6635  6641  6646  6652  6658  6664  6670  6676  6682 

6688  6694  6700  6705  6711  6717  6723  6729  6735  6741 

6747  6753  6759  6764  6770  6776  6782  6788  6794  6800 

6806  6812  6817  6823  6829  6835  6841  6847  6853  6859 

6864  6870  6876  6882  6888  6894  6900  6906  6911  6917 

86923  86929  86935  86941  86947  86953  86958  86964  86970  86976 

6982  6988  6994  6999  7005  7011  7017  7023  7029  7035 

7040  7046  7052  7058  7064  7070  7075  7081  7087  7093 

7099  7105  7111  7116  7122  7128  7134  7140  7146  7151 

7157  7163  7169  7175  7181  7186  7192  7198  7204  7210 

7216  7221  -7227  7233  7239  7245  7251  7256  7262  7268 

7274  7280  7286  7291  7297  7303  7309  7315  7320  7326 

7332  7338  7344  7349  7355  7361  7367  7373  7379  7384 

7390  7396  7402  7408  7413  7419  7425  7431  7437  7442 

7448  7454  7460  7466  7471  7477  7483  7489  7495  7500 

87506  87512  87518  87523  87529  87535  87541  87547  87552  87558 


130  LOGARITHMS  OF  NUMBERS 


NO        1234567        8       9 


750 
1 

2 
3 

4 
5 
6 
7 
8 
9 

760 

1 
2 
3 
4 
5 
6 
7 
8 
9 

770 
1 

2 
3 
4 
5 
6 
7 
8 
9 

780 
1 
2 
3 
4 
5 
6 
7 
8 
9 

790 
1 
2 
3 
4 
5 
6 
7 


800 


87506  87512  87518  87523  87529  87535  87541  87547  87552  87558 

7564  7570  7576  7581  7587  7593  7599  7604  7610  7610 

7622  7628  7633  7639  7645  7651  7656  7662  7668  7674 

7679  7685  7691  7697  7703  7708  7714  7720  7726  7731 

7737  7743  7749  7754  7760  7766  7772  7777  7783  7789 

7795  7800  7806  7812  7818  7823  7829  7835  7841  7846 

7852  7858  7864  7869  7875  7881  7887  7892  7898  7904 

7910  7915  7921  7927  7933  7938  7944  7950  7955  7961 

7967  7973  7978  7984  7990  7996  8001  8007  8013  8018 

8024  8030  8036  8041  8047  8053  8058  8064  8070  8076 

88081  88087  88093  88098  88104  88110  88116  88121  88127  88133 

8138  8144  8150  8156  8161  8167  8173  8178  8184  8190 

8195  8201  8207  8213  8218  8224  8230  8235  8241  8247 

8252  8258  8264  8270  8275  8281  8287  8292  8298  8304 

8309  8315  8321  8326  8332  8338  8343  8349  8355  8360 

8366  8372  8377  8383  8389  8395  8400  8406  8412  8417 

8423  8429  8434  8440  8446  8451  8457  8463  8468  8474 

8480  8485  8491  8497  8502  8508  8513  8519  8525  8530 

8536  8542  8547  8553  8559  8564  8570  8576  8581  8587 

8593  8598  8604  8610  8615  8621  8627  8632  8638  8643 

88649  88655  88660  88666  88672  88677  88683  88689  88694  88700 

8705  8711  8717  8722  8728  8734  8739  8745  8750  8756 

8762  8767  8773  8779  8784  8790  8795  8801  8807  8812 

8818  8824  8829  8835  8840  8846  8852  8857  8863  8868 

8874  8880  8885  8891  8897  8902  8908  8913  8919  8925 

8930  8936  8941  8947  8953  8958  8964  8969  8975  8981 

8986  8992  8997  9003  9009  9014  9020  9025  9031  9037 

9042  9048  9053  9059  9064  9070  9076  9081  9087  9092 

9098  9104  9109  9115  9120  9126  9131  9137  9143  9148 

9154  9159  9165  9170  9176  9182  9187  9193  9198  9204 

89209  89215  89221  89226  89232  89237  89243  89248  89254  89260 

9265  9271  9276  9282  9287  9293  9298  9304  9310  9315 

9321  9326  9332  9337  9343  9348  9354  9360  9365  9371 

9376  9382  9387  9393  9398  9404  9409  9415  9421  9426 

9432  9437  9443  9448  9454  9459  9465  9470  9476  9481 

9487  9492  9498  9504  9509  9515  9520  9526  9531  9537 

9542  9548  9553  9559  9564  9570  9575  9581  9586  9592 

9597  9603  9609  9614  9620  9625  9631  9636  9642  9647 

9653  9658  9664  9669  9675  9680  9686  9691  9697  9702 

9708  9713  9719  9724  9730  9735  9741  9746  9752  9757 

89763  89768  89774  89779  89785  89790  89796  89801  89807  89812 

9818  9823  9829  9834  9840  9845  9851  9856  9862  9867 

9873  9878  9883  9889  9894  9900  9905  9911  9916  9922 

9927  9933  9938  9944  9949  9955  9960  9966  9971  9977 

9982  9988  9993  9998  90004  90009  90015  90020  90026  90031 

90037  90042  90048  90053  0059  0064  0069  0075  0080  0086 

0091  0097  0102  0108  0113  0119  0124  0129  0135  0140 

0146  0151  0157  0162  0168  0173  0179  0184  0189  0195 

0200  0206  0211  0217  0222  0227  0233  0238  0244  0249 

0255  0260  0266  0271  0276  0282  0287  0293  0298  0304 

90309  90314  90320  90325  90331  90336  90342  90347  90352  90358 


LOGARITHMS  OF  NUMBERS  131 


NO        12        34        567        89 


800 

1 
2 
3 
4 
5 


9 

810 

1 
2 
3 

4 
5 
6 

7 
8 


820 
1 
2 
3 
4 
5 
6 
7 
8 


830 

1 
2 
3 
4 
5 
6 
7 


340 

1 
2 
3 

4 
5 
6 

7 


90309  90314  90320  90325  90331  90336  90342  90347  90352  90358 

0363  0369  0374  0380  0385  0390  0396  0401  0407  0412 

0417  0423  0428  0434  0439  0445  0450  0455  0461  0466 

0472  0477  0482  0488  0493  0499  0504  0509  0515  0520 

0526  0531  0536  0542  0547  0553  0558  0563  0569  0574 

0580  0585  0590  0596  0601  0607  0612  0617  0623  0628 

0634  0639  0644  0650  0655  0660  0666  0671  0677  0682 

0687  0693  0698  0703  0709  0714  0720  0725  0730  0736 

0741  0747  0752  0757  0763  0768  0773  0779  0784  0789 

0795  0800  0806  0811  0816  0822  0827  0832  0838  0843 

90849  90854  90859  90865  90870  90875  90881  90886  90891  90897 

0902  0907  0913  0918  0924  0929  0934  0940  0945  0950 

0956  0961  0966  0972  0977  0982  0988  0993  0998  1004 

1009  1014  1020  1025  1030  1036  1041  1046  1052  1057 

1062  1068  1073  1078  1084  1089  1094  1100  1105  1110 

1116  1121  1126  1132  1137  1142  1148  1153  1158  1164 

1169  1174  1180  1185-  1190  1196  1201  1206  1212  1217 

1222  1228  1233  1238  1243  1249  1254  1259  1265  1270 

1275  1281  1286  1291  1297  1302  1307  1312  1318  1323 

1328  1334  1339  1344  1350  1355  1360  1365  1371  1376 

91381  91387  91392  91397  91403  91408  91413  91418  91424  91429 

1434  1440  1445  1450  1455  1461  1466  1471  1477  1482 

1487  1492  1498  1503  1508  1514  1519  1524  1529  1535 

1540  1545  1551  1556  1561  1566  1572  1577  1582  1587 

1593  1598  1603  1609  1614  1619  1624  1630  1635  1640 

1645  1651  1656  1661  1666  1672  1677  1682  1687  1693 

1698  1703  1709  1714  1719  1724  1730  1735  1740  1745 

1751  1756  1761  1766  1772  1777  1782  1787  1793  1798 

1803  1808  1814  1819  1824  1829  1834  1840  1845  1850 

1855  1861  1866  1871  1876  1882  1887  1892  1897  1903 

91908  91913  91918  91924  91929  91934  91939  91944  91950  91955 

1960  1965  1971  1976  1981  1986  1991  1997  2002  2007 

2012  2018  2023  2028  2033  2038  2044  2049  2054  2059 

2065  2070  2075  2080  2085  2091  2096  2101  2106  2111 

2117  2122  2127  2132  2137  2143  2148  2153  2158  2163 

2169  2174  2179  2184  2189  2195  2200  2205  2210  2215 

2221  2226  2231  2236  2241  2247  2252  2257  2262  2267 

2273  2278  2283  2288  2293  2298  2304  2309  2314  2319 

2324  2330  2335  2340  2345  2350  2355  2361  2366  2371 

2376  2381  2387  2392  2397  2402  2407  2412  2418  2423 

92428  92433  92438  92443  92449  92454  92459  92464  92469  92474 

2480  2485  2490  2495  2500  2505  2511  2516  2521  2526 

2531  2536  2542  2547  2552  2557  2562  2567  2572  2578 

2583  2588  2593  2598  2603  2609  2614  2619  2624  2629 

2634  2639  2645  2650  2655  2660  2665  2670  2675  2681 

2686  2691  2696  2701  2706  2711  2716  2722  2727  2732 

2737  2742  2747  2752  2758  2763  2768  2773  2778  2783 

2788  2793  2799  2804  2809  2814  2819  2824  2829  2834 

2840  2845  2850  2855  2860  2865  2870  2875  2881  2886 

2891  2896  2901  2906  2911  2916  2921  2927  2932  2937 

850  92942  92947  92952  92957  92962  92967  92973  92978  92983  92988 


LOGARITHMS  OF  NUMBERS 


850 
1 
2 
3 
4 
5 
6 
7 
8 
9 

860 

1 
2 
3 
4 
5 
6 
7 
8 
9 

870 
1 
2 
3 
4 
5 
6 
7 


880 
1 
2 
3 
4 
5 
6 
7 
8 


890 
1 
2 
3 

4 
5 


900 


5678 


92942  92947  02952  92957  92962  92967  92973  92978  92983  92988 

2993  2998  3003  3008  3013  3018  3024  3029  3034  3039 

3044  3049  3054  3059  3064  3069  3075  3080  3085  3090 

3095  3100  3105  3110  3115  3120  3125  3131  3136  3141 

3146  3151  3156  3161  3166  3171  3176  3181  3186  3192 

3197  3202  3207  3212  3217  3222  3227  3232  3237  3242 

3247  3252  3258  3263  3268  3273  3278  3283  3288  3293 

3298  3303  3308  3313  3318  3323  3328  3334  3339  3344 

3349  3354  3359  3364  3369  3374  3379  3384  3389  3394 

3399  3404  3409  3414  3420  3425  3430  3435  3440  3445 

93450  93455  93460  93465  93470  93475  93480  93485  93490  93495 

3500  3505  3510  3515  3520  3526  3531  3536  3541  3546 

3551  3556  3561  3566  3571  3576  3581  3586  3591  3596 

3601  3606  3611  3616  3621  3626 .  3631  3636  3641  3646 

3651  3656  3661  3666  3671  3676  3682  3687  3692  3697 

3702  3707  3712  3717  3722  3727  3732  3737  3742  3747 

3752  3757  3762  3767  3772  3777  3782  3787  3792  3797 

3802  3807  3812  3817  3822  3827  3832  3837  3842  3847 

3852  3857  3862  3867  3872  3877  3882  3887  3892  3897 

3902  3907  3912  3917  3922  3927  3932  3937  3942  3947 

93952  93957  93962  93967  93972  93977  93982  93987  93992  93997 

4002  4007  4012  4017  4022  4027  4032  4037  4042  4047 

4052  4057  4062  4067  4072  4077  4082  4086  4091  4096 

4101  4106  4111  4116  4121  4126  4131  4136  4141  4146 

4151  4156  4161  4166  4171  4176  4181  4186  4191  4196 

4201  4206  4211  4216  4221  4226  4231  4236  4240  4245 

4250  4255  4260  4265  4270  4275  4280  4285  4290  4295 

4300  4305  4310  4315  4320  4325  4330  4335  4340  4345 

4349  4354  4359  4364  4369  4374  4379  4384  4389  4394 

4399  4404  4409  4414  4419  4424  4429  4433  4438  4443 

94448  94453  94458  94463  94468  94473  94478  94483  94488  94493 

4498  4503  4507  4512  4517  4522  4527  4532  4537  4542 

4547  4552  4557  4562  4567  4571  4576  4581  4586  4591 

4596  4601  4606  4611  4616  4621  4626  4630  4635  4640 

4645  4650  4655  4660  4665  4670  4675  4680  4685  4689 

4694  4699  4704  4709  4714  4719  4724  4729  4734  4738 

4743  4748  4753  4758  4763  4768  4773  4778  4783  4787 

4792  4797  4802  4807  4812  4817  4822  4827  4832  4836 

4841  4846  4851  4856  4861  4866  4871  4876  4880  .4885 

4890  4895  4900  4905  4910  4915  4919  4924  4929  4934 

94939  94944  94949  94954  94959  94963  94968  94973  94978  94983 

4988  4993  4998  5002  5007  5012  5017  5022  5027  5032 

5036  5041  5046  5051  5056  5061  5066  5071  5075  5080 

5085  5090  5095  5100  5105  5109  5114  5119  5124  5129 

5134  5139  5143  5148  5153  5158  5163  5168  5173  5177 

5182  5187  5192  5197  5202  5207  5211  5216  5221  5226 

5231  5236  5240  5245  5250  5255  5260  5265  5270  5274 

5279  5284  5289  5294  5299  5303  5308  5313  5318  5323 

5328  5332  5337  5342  5347  5352  5357  5361  5366  5371 

5376  5381  5386  5390  5395  5400  5405  5410  5415  5419 

95424  95429  95434  95439  95444  95448  95453  95458  95463  95468 


LOGARITHMS  OF  NUMBERS  133 


0123456789 


900  95424  95429  95434  95439  95444  95448  95453  95458  95463  95468 

1  5472  5477  5482  5487  5492  5497  5501  5506  5511  5516 

2  5521  5525  5530  5535  5540  5545  5550  5554  5559  5564 

3  5569  5574  5578  5583  5588  5593  5598  5602  5607  5612 

4  5617  5622  5626  5631  5636  5641  5646  5650  5655  5660 

5  5665  5670  5674  5679  5684  5689  5694  5698  5703  5708 

6  5713  5718  5722  5727  5732  5737  5742  5746  5751  5756 

7  5761  5766  5770  5775  5780  5785  5789  5794  5799  5804 
5809  5813  5818  5823  5828  5832  5837  5842  5847  5852 
5856  5861  5866  5871  5875  5880  5885  5890  5895  5899 

910  95904  95909  95914  95918  95923  95928  95933  95988*95942  95947 

1  5952  5957  5961  5966  5971  5976  5980  5985  5990  5995 

2  I  5999  6004  6009  6014  6019  6023  6028  6033  6038  6042 

3  6047  6052  6057  6061  6066  6071  6076  6080  6085  6090 


9 

920 

1 
2 
3 

4 
5 
6 
7 
8 
9 

980 

1 
2 
3 

4 
5 
6 
7 
8 
9 

940 

1 
2 
3 
4 
5 
6 
7 
8 
9 

950 


6095  6099  6104  6109  6114  6118  6123  6128  6133  6137 

6142  6147  6152  6156  6161  6166  6171  6175  6180  6185 

6190  6194  6199  6204  6209  6213  6218  6223  6227  6232 

6237  6242  6246  6251  6256  6261  6265  6270  6275  6280 

6284  6289  6294  6298  6303  6308  6313  6317  6322  6327 

6332  6336  6341  6346  6350  6355  6360  6365  6369  6374 

96379  96384  96388  96393  96398  96402  96407  96412  96417  96421 

6426  6431  6435  6440  6445  6450  6454  6459  6464  6468 

6473  6478  6483  6487  6492  6497  6501  6506  6511  6515 

6520  6525  6530  6534  6539  6544  6548  6553  6558  6562 

6567  6572  6577  6581  6586  6591  6595  6600  6605,6609 

6614  6619  6624  6628  6633  6638  6642  6647  6652  6656 

6661  6666  6670  6675  6680  6685  6689  6694  6699  6703 

6708  6713  6717  6722  6727  6731  6736  6741  6745  6750 

6755  6759  6764  6769  6774  6778  6783  6788  6792  6797 

6802  6806  6811  6816  6820  6825  6830  6834  6839  6844 

96848  96853  96858  96862  96867  96872  96876  96881  96886  96890 

6895  6900  6904  6909  6914  6918  6923  6928  6932  6937 

6942  6946  6951  6956  6960  6965  6970  6974  6979  6984 

6988  6993  6997  7002  7007  7011  7016  7021  7025  7030 

7035  7039  7044  7049  7053  7058  7063  7067  7072  7077 

7081  7086  7090  7095  7100  7104  7109  7114  7118  7123 

7128  7132  7137  7142  7146  7151  7155  7160  7165  7169 

7174  7179  7183  7188  7192  7197  7202  7206  7211  7216 

7220  7225  7230  7234  7239  7243  7248  7253  7257  7262 

7267  7271  7276  7280  7285  7290  7294  7299  7304  7308 

97313  97317  97322  97327  97331  97336  97340  97345  97350  97354  I 

7359  7364  7368  7373  7377  7382  7387  7391  7396  7400  ' 

7405  7410  7414  7419  7424  7428  7433  7437  7442  7447 

7451  7456  7460  7465  7470  7474  7479  7483  7488  7493 

7497  7502  7506  7511  7516  7520  7525  7529  7534  7539 

7543  7548  7552  7557  7562  7566  7571  7575  7580  7585 

7589  7594  7598  7603  7607  7612  7617  7621  7626  7630 

7635  7640  7644  7649  7653  7658  7663  7667  7672  7676 

7681  7685  7690  7695  7699  7704  7708  7713  7717  7722 

7727  7731  7736  7740  7745.  7749  7754  7759  7763  7768 

97772  97777  97782  97786  97791  97795  97800  97804  97809  97813 


134  LOGARITHMS  OF  NUMBERS 


O12345678 


950 

1 
2 
3 

4 
5 
6 

7 


960 


970 

1 
2 

3 
4 
5 
6 

7 


980 

1 
2 
3 
4 
5 
6 
7 
8 
9 

990 
1 
2 
3 
4 
5 
f? 
7 
8 
9 


97772  97777  97782  97786  97791  97795  97800  97804  97809  97813 

7818  7823  7827  7832  7836  7841  7845  7850  7855  7859 

7864  7868  7873  7877  7882  7886  7891  7896  7900  7905 

7909  7914  7918  7923  7928  7932  7937  7941  7946  7950 

7955  7959  7964  7968  7973  7978  7982  7987  7991  7996 

8000  8005  8009  8014  8019  8023  8028  8032  8037  8041 

8046  8050  8055  8059  8064  8068  8073  8078  8082  8087 

8091  8096  8100  8105  8109  8114  8118  8123  8127  8132 

8137  8141  8146  8150  8155  8159  8164  8168  8173  8177 

8182  8186-8191  8195  8200  8204  8209  8214  8218  8223 

.98227  98232  98236  98241  98245  98250  98254  98259  98263  98268 

8272  8277  8281  8286  8290  8295  8299  8304  8308  8313 

8318  8322  8327  8331  8336  8340  8345  8349  8354  8358 

8363  8367  8372  8376  8381  8385  8390  8394  8399  8403 

8408  8412  8417  8421  8426  8430  8435  8439  8444  8448 

8453  8457  8462  8466  8471  8475  8480  8484  8489  8493 

8498  8502  8507  8511  8516  8520  8525  8529  8534  8538 

8543  8547  8552  8556  8561  8565  8570  8574  8579  8583 

8588  8592  8597  8601  8605  8610  8614  8619  8623  8628 

8632  8637  8641  8646  8650  8655  8659  8664  8668  8673 

98677  98682  98686  98691  98695  98700  98704  98709  98713  98717 

8722  8726  8731  8735  8740  8744  8749  8753  8758  8762' 

8767  8771  8776  8780  8784  8789  8793  8798  8802  8807 

8811  8816  8820  8825  8829  8834  8838  8843  8847  8851 

8856  8860  8865  8869  8874  8878  8883  8887  8892  8896 

8900  8905  8909  8914  8918  8923  8927  8932  8936  8941 

8945  8949  8954  8958  8963  8967  8972  8976  8981  8985 

8989  8994  8998  9003  9007  9012  9016  9021  9025  9029 

9034  9038  9043  9047  9052  9056  9061  9065  9069  9074 

9078  9083  9087  9092  9096  9100  9105  9109  9114  9118 

99123  99127  99131  99136  99140  99145  99149  99154  99158  99162 

9167  9171  9176  9180  9185  9189  9193  9198  9202  9207 

9211  9216  9220  9224  9229  9233  9238  9242  9247  9251 

9255  9260  9264  9269  9273  9277  9282  9286  9291  9295 

9300  9304  9308  9313  9317  9322  9326  9330  9335  9339 

9344  9348  9352  9357  9361  9366  9370  9374  9379  9383 

9388  9392  9396  9401  9405  9410  9414  9419  9423  9427 

9432  9436  9441  9445  9449  9454  9458  9463  9467  9471 

9476  9480  9484  9489  9493  9498  9502  9506  9511  9515 

9520  9524  9528  9533  9537  9542  9546  9550  9555  .  9559 

99564  99568  99572  99577  99581  99585  99590  99594  99599  99603 

9607  9612  9616  9621  9625  9629  9634  9638  9642  9647 

9651  9656  9660  9664  9669  9673  9677  9682  9686  9691 

9699  9704  9708  9712  9717  9721  9726  9730  9734 

9739  9743  9747  9752  9756  9760  9765  9769  9774  9778 

9782  9787  9791  9795  9800  9804  9808  9813  9817  9822 

9826  9830  9835  9839  9843  9848  9852  9856  9861  9865 

9870  9874  9878  9883  9887  9891  9896  9900  9904  9909 

9913  9917  9922  9926  9930  9935  9939  9944  9948  9952 

9957  9961  9965  9970  9974  9978  9983  9987  9991  9996 


000  00000  00004  00009  00013  00017  00022  00026  00030  00035  00039 


LOGARITHMIC  SINES  AND  COSINES 


o° 


1° 


Sine       Cosine 


Sine 


Cosine 


Sine       Cosine 


— oo 

6.46373 

76476 

94085 

7.06579 

16270 

24188 


41797 

7.46373 
50512 
54291 

57767 
60985 


66784 
69417 
71'JOO 
74248 

7.76475 
78594 
80615 
82545 
84393 
86166 
87870 
89509 
91088 
92612 

7.94084 
95508 
96887 


99520 
8.00779 
02002 
03192 
04350 
05478 

8.06578 
07650 
08696 
09718 
10717 
11693 
12647 
13581 
14495 
15391 

8.16268 
17128 
17971 
18798 
19610 
20407 
21189 
21958 
22713 
23456 
24186 


10.00000 
00000 
00000 
00000 
00000 
00000 
00000 
00000 
00000 
00000 

10.00000 
00000 
00000 
00000 
00000 
00000 
00000 
9.99999 
99999 
99999 


99999 


.  99998 
99998 
99998 


99998 


99997 
99997 
99997 

.99997 
99997 
99997 
99997 
99996 
99996 
99996 


99996 


.99995 
99995 
99995 
99995 
99995 
99994 
99994 


27661 
28324 


29621 


5. 30879 
31495 
32103 
32702 


33875 
34450 
35018 
35578 
36131 

.36678 
37217 
37750 
38276 


39310 


40320 
40816 
41307 

.41792 
42272 
42746 
43216 
43680 
44139 
44594 
45044 
45489 
45930 


48485 
48896 
49304 
49708 
50108 

.50504 
50897 
51287 
51673 
52055 
52434 
52810 
53183 
53552 
53919 
54282 


.24186 
24903 
25609 
26304  99993 


99992 


99992 
99992 


99990 
99990 


99987 
99987 

99987 


9.99985 
9998E 


99984 
99984 
99983 


99983 


99982 


8.46366        9.99982 
46799 
47226 
47650  99981 


99979 
99979 


9.99978 
99977 
99977 
99977 


99975 
99974 
99974 
99974 


8.54282 
54642 
54999 
55354 
55705 
56054 
56400 
56743 
57084 
57421 


58419 
58747 
59072 
59395 
59715 


60662 

8.60973 
61282 
61589 
61894 
62196 
62497 
62795 


63385 
63678 

8.63968 
64256 
64543 
64827 
65110 
65391 
65670 
65947 


66497 
8.60769 


67575 
67841 
68104 
68367 
68627 


69144 

8.69400 
69654 
69907 
70159 
70409 
70658 
70905 
71151 
71395 
71638 
71880 


9.99974 
99973 
99973 
99972 
99972 
69971 
99971 
99970 


9.99991   8.57757   9 


99967 


99966 
99966 


99964 

9.99964 
99963 


99962 


99960 
99959 

.99959 


99958 
99957 
99956 
99956 
99955 
99955 
99954 
9!)954 

.99953 
99952 
99952 
99951 
99951 
99950 
99949 


9.99947 


99945 


99942 
99942 


Cosine       Sine 
89° 


Cosine 


Sine 


Cosine 


Sine 


88° 


130 


LOGARITHMIC  SINES  AND   COSINES 


t 

3° 

4° 

6° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

8.71880 

9.99940 

8.84358 

9.99894 

8.94030 

9.99834 

60 

1 

72120 

99940 

84539 

99893 

94174 

99833 

59 

2 

72359 

99939 

84718 

99892 

94317 

99832 

58 

3 

72597 

!>9938 

84897 

99891 

944(51 

99831 

57 

4 

72834 

99938 

85075 

99891 

94603 

99830 

56 

5 

.  73069 

99937 

85252 

99890 

94746 

99829 

55 

6 

73303 

99936 

85429 

99889 

94887 

99828 

54 

7 

73535 

99936 

85605 

99888 

95029 

99827 

53 

8 

73767 

99935 

85780 

99887 

95170 

99825 

52 

9 

73997 

99934 

85955 

99886 

95310 

99824 

51 

10 

8.74226 

9.99934 

8.86128 

9.99885 

8.95450 

9.99823 

50 

11 

74454 

99933 

86301 

99884 

95589 

99822    49 

12 

74680 

99932 

86474 

99883 

95728 

99821    48 

13 

74906 

99932 

86645 

99882 

95867 

99820    47 

14 

75130 

99931 

86816 

99881 

9(5005 

99819 

46 

15 

75353 

99930 

86987 

99880 

96143 

99817 

45 

16 

75575 

99929 

87156 

99879 

96280 

99816 

44 

17 

75795 

99929 

87325 

99879 

96417 

99815 

43 

18 

76015 

99928 

87494 

99878 

96553 

99814 

42 

19 

76234 

99927 

8766  1 

99877 

96689 

99813 

41 

20 

8.76451 

9.99926 

8.87829 

9.99876 

8.96825 

9.99812 

40 

21 

76667 

99926 

87995 

99875 

96960 

99810 

39 

22 

76883 

99925 

88161 

99874 

97095 

99809 

38 

23 

77097 

99924 

88326 

99873 

97229 

93808 

37 

24 

77310 

99923 

88490 

99872 

97363 

99807 

36 

25 

77522 

99923 

88654 

99871 

97496 

99806 

35 

26 

77733 

99922 

88817 

99870 

97629 

99804 

34 

27 

77943 

99921 

88980 

99869 

97762 

90803 

33 

28 

78152 

99920 

89142 

99868 

97894 

99802 

33 

29 

78360 

99920 

89304 

99867 

98026 

99801 

31 

30 

8.78568 

9.99919 

8.89464 

9.99866 

8.98157 

9.99800 

30 

31 

78774 

99918 

89625 

9981)5 

98288 

99798 

29 

32 

78979 

99917 

89784 

99864 

98419 

99797 

28 

33 

79183 

99917 

89943 

99863 

98549 

99796 

27 

34 

79386 

99916 

P0102 

99862 

98679 

99795 

26 

35 

79588 

99915 

90260 

99861 

98808 

99793 

25 

36 

79789 

99914 

90417 

99860 

98937 

99792 

24 

87 

79990 

99913 

'  90574 

99859 

99066 

99791 

23 

38 

80189 

99913 

90730 

99858 

99194 

99790 

22 

39 

80388 

99912 

90885 

99857 

99322 

99788 

21 

40 

8.80585 

9.99911 

8.91040 

9.99856 

8.99450 

9.99787 

20 

41 

8078-2 

99910 

91195 

99855 

99577 

99786 

19 

42 

80978 

99909 

91349 

99854 

99704 

99785 

18 

43 

81173 

99909 

91502 

99853 

99830 

99783 

17 

44 

81367 

99908 

91655 

99852 

99956 

99782 

16 

45 

81560 

99907 

91807 

99851 

9.00082 

99781 

15 

46 

81752 

99906 

91959 

99850 

00207 

99780 

14 

47 

81944 

99905 

92110 

99848 

00332 

99778 

13 

48 

82134 

99904 

92261 

99847 

00456 

99777 

12 

49 

82324 

99904 

92411 

99846 

00581 

99776 

11 

50 

8.82513 

9.99903 

8.92561 

9.99845 

9.00704 

9.99775 

10 

51 

82701 

99902 

92710 

99844 

00828 

99773 

9 

52 

82888 

99901 

92859 

99843 

00951 

99772 

8 

53 

83075 

99900 

93007 

99842 

01074 

99771 

7 

54 

83261 

9'.»899 

93154 

99S41 

01196 

99769 

6 

55 

83446 

99898 

93301 

99840 

01318 

99768 

5 

56 

83630 

99898 

93448 

99839 

01440 

99767 

4 

57 

83813 

99897 

93594 

99838 

01561 

99705 

3 

58 

83996 

99896 

93740 

99837 

01682 

99764 

2 

59 

84177 

99895 

93885 

99836 

01803 

99763 

1 

60 

84358 

99894 

94030 

99834 

01923 

99761 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

86° 

- 

85° 

84° 

LOGARITHMIC  SINES  AND  COSINES 


Sine       Cosine 


Sine 


Cosine 


Sine         Cosine 


9.01923 
02043 
02163 
02283 

.  02402 
02520 
02639 
02757 
02874 
02992 

9.03109 
03226 
03342 
03 158 
03574 
0  !690 
03805 
03920 
04034 
04149 

9.04262 
04376 
04490 
04603 
04715 
04828 
04940 
05052 
05164 
05275 

9.05386 
05497 
05607 
05717 
05827 
05937 
06046 
06155 
06264 
06372 

9.06481 
06589 


06804 


07018 
07124 
07231 
07337 
07442 

.07548 
07653 
07758 

07863 
07968 
08072 
08176 
08280 
08383 


08589 


9.99761 
99760 
99759 
99757 
99756 
99755 
99753 
99752 
99751 
99749 

9.99748 
99747 
99745 
99744 
99742 
99741 
99740 


99737 
99736 

9.99734 
99733 
99731 
99730 
99728 
99727 
99726 
99724 
99723 
99721 

9.99720 
99718 
99717 
99716 
99714 
9CC13 
99711 
99710 


99707 


9.99705 
99704 


99701 
99699 


99692 


99687 


99684 


99678 
99677 
99675 


08795 


09101 
09202 


09506 


09707 
09807 
09907 
10006 
10106 
10205 
i0304 
10402 
10501 

9.10599 
10697 
10795 


10990 
11087 
11184 
11281 
11377 
11474 

9.11570 
11666 
11761 
11857 
11952 
12047 
12142 
12236 
12331 
12425 

9.12519 
12612 
12706 
12799 


13078 
13171 
13263 
13355 

9.13447 
13539 
13630 
13722 
13813 
18904 


14085 
14175 


9.99675 
99674 


99666 


99663 
99661 


99656 


99653 
99651 
99650 
99648 
99647 
99645 

.99643 


99637 
99635 
99633 


99625 


99615 


99612 
&. 99610 


99595 
9.99593 


99589 


99586 
99584 


14356 


99581 
99579 
99577 
99575 


9.14356 
14445 
14535 
14624 
14714 
14803 


15069 
15157 

9.15245 
15333 
15421 
15508 


15683 
15770 
15857 
15944 


9.16116 
16203 
16289 
16374 
16460 
16545 
16631 
16716 
16801 


17055 
17139 
17223 
]7807 
17391 
17474 
17558 
17641 
17724 

.17807 
17890 
17973 
18055 
18137 
18220 
18302 


18465 
18547 

9.18628 
18709 
18790 
18871 
18952 
19033 
19113 
19193 
19273 
19358 


Cosine         Sine         Cosine 

..:  88° 


Sine         Cosine 


9.99575 
99574 
99572 
99570 
99568 


99565 


99561 


9.99557 
99556 
99554 
99552 
99550 
99548 
99546 
99545 
99543 
99541 

9.99539 
99537 
99535 
99533 
99532 
99530 
89528 
99526 
99524 
99522 

9.99520 
99518 
99517 
995  J  5 
99513 
99511 
99509 
99507 
99505 


9.99501 


99492 
99490 


99486 
99484 

9.99482 


99478 
99476 
99474 
99472 
99470 
99468 


99464 
99462 


Sine 


81' 


138 


LOGARITHMIC   SINES  AND   COSINES 


/ 

9° 

10° 

11° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.19433 

9.99462 

9.23967 

9.99335 

9.28060 

9.99195 

60 

1 

19513 

99460 

24089 

99333 

28125 

99192 

59 

2 

19592 

99458 

24110 

99331 

28190 

99190 

58 

3 

19672 

99456 

24181 

99328 

28254 

99187 

57 

4 

19751 

99454 

24253 

99326 

28319 

99185 

56 

5 

19830 

99452 

24324 

99324 

28384 

99183 

55 

6 

19909 

99450 

24395 

99322 

28448 

99180 

54 

7 

19988 

99448 

24466 

99319 

28512 

99177 

53 

8 

20067 

99446 

24536 

99317 

28577 

99175 

52 

9 

20145 

99444 

24607 

99315 

28641 

99172 

51 

10 

9.20223 

9.99442 

9.24677 

9.99313 

9.28705 

9.99170 

50 

11 

20302 

99440 

24748 

99310 

28769 

99167 

49 

12 

20380 

99438 

24818 

99308 

28833 

99165 

48 

13 

20458 

99436 

24888 

99306 

28896 

99162 

47 

14 

20535 

99434 

24958 

99304 

28960 

99160 

46 

15 

20613 

99432 

25028 

99301 

29024 

99157 

45 

16 

20691 

99429 

25098 

99299 

29087 

99155  ' 

44 

17 

20768 

99427 

25168 

99297 

29150 

99152 

43 

18 

20845 

99425 

25237 

99294 

29214 

99150 

42 

19 

20922 

99423 

25307 

99-292 

29277 

99147 

41 

20 

9.20999 

9.99421 

9.25376 

9.99290 

9.29340 

9.99145 

40 

21. 

21076 

99419 

25445 

99288 

29403 

99142 

39 

22 

21153 

99417 

25514 

99285 

29466 

99140 

38 

23 

21229 

99415 

25583 

99283 

29529 

99137 

37 

24 

21306 

99413 

25652 

99281 

29591 

99135 

36 

25 

21382 

99411 

25721 

99278 

29654 

99132 

35 

23 

21458 

99409 

25790 

99276 

29716 

99130 

34 

27 

21534 

99407 

25858 

99274 

29779 

99127 

33 

28 

21610 

99404 

25927 

99271 

29841 

99124 

32 

29 

21685 

99402 

25995 

99269 

29903 

99122 

31 

30 

9.21761 

9.99400 

9.260C3 

9.99267 

9.29966 

9.99119 

30 

31 

21836 

99398 

26131 

99264 

30028 

99117 

29 

32 

21912 

99396 

26199 

99262 

30090 

99114 

28 

33 

21987 

99394 

26267 

99260 

30151 

99112 

27 

34 

22062 

99392 

26335 

99257 

30213 

99109 

26 

as 

22137 

99390 

26403 

99255 

30275 

99106 

25 

36 

22211 

99388 

26470 

99252 

30336 

99104 

24 

37 

22286 

99385 

26538 

99250 

30398 

99001 

23 

38 

22361 

99383 

26605 

99248 

30459 

99099 

22 

39 

22435 

99381 

26672 

99245 

30521 

99096 

21 

40 

9.22509 

9.99379 

9.26739 

9.99243 

9.30582 

9.99093 

20 

41 

22588 

99377 

26806 

99241 

30643 

99091 

19 

42 

22657 

99375 

26873 

99238 

30704 

99088 

18 

43 

22731 

99372 

26940 

99236 

30765 

99086 

17 

44 

22805 

99370 

27007 

99233 

30826 

99083 

16 

45 

22878 

99368 

27073 

99231 

30887 

99080 

15 

46 

22952 

99366 

27140 

99229 

30947 

99078 

14 

47 

23025 

99364 

27206 

99226 

31008 

99075 

13 

48 

23098 

99362 

27273 

99224 

31068 

99072 

12 

49 

23171 

99359 

27339 

99221 

31129 

99070 

11 

50 

9.23244 

9.99357 

9.27405 

9.99^19 

9.31189 

9/J9067 

10 

51 

23317 

99355 

27471 

99217 

81250 

99064 

9 

52 

23390 

99353 

27537 

99214 

31310 

99062 

8 

53 

23462 

99351 

27602 

99212 

31370 

99059 

7 

54 

23535 

99348 

27668 

99209 

31430 

99056 

6 

55 

23607 

99346 

27734 

99207 

31490 

99054 

5 

56 

23679 

99344 

27799 

99204 

31549 

99051 

4 

57 

23752 

9934? 

27864 

99202 

31609 

99048 

3 

58 

23823 

99340 

27930 

99200 

31669 

99046 

2 

59 

23895 

99337 

27995 

99197 

31728 

99043 

1 

60 

23967 

99335 

28060 

99195 

31788 

99040 

0 

1 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

4 

80° 

79° 

78° 

LOGARITHMIC  SINES  AND  COSINES 


12° 


13= 


Sine    Cosine 


9.31788 
31847 
31907 
31966 
320:25 
32084 
32143 
32202 
32261 
32319 

9.32378 
38437 
32495 
32553 
32612 


32786 
32844 
32902 

9.32960 
33018 
'  33075 
33133 
33190 
33248 
33305 
33362 
33420 
33477 

9.33534 
3359  J 
33647 
33704 
33761 
33818 
33874 
33931 
33987 
34043 

9.34100 
34156 
34212 


34324 


35154 
35209 


Sine 


Cosine 


Sine 


Cosine 


99030 
99027 
99024 
99022 
99019 
9a016 

9.99013 
99011 
99008 
9900:5 
99002 
99000 


98991 


98975 
98972 


98967 
98964 


98953 


98947 
98944 


9.98930 
.  98927 


98910 


34436 
34491 
34547 
34602  98904 

9.34658   9, 
34713   ' 
34769 
34824 
34879 
34934 
34989 
35014 


98898 


98875 
98872 


.35209 
35263 
35318 
35373 
35427 
35481 
35536 
35590 
35644 
35698 

.35752 
35806 
35860 
35914 


36022 
36075 
36129 
36182 


36449 
36502 
36555 


36660 
36713 
36766 


36871 
36924 


37028 
37081 
37133 
37185 
37237 
37289 

9.37341 


37445 
37497 
37549 
37600 
37652 
37703 
37755 
37806 

.37858 
37909 
37960 
38011 
38062 
38113 
38164 
38215 
38266 
38317 


9.98872 


1)8867 
98864 


98858 
98855 


98849 
98346 


9.98843 

98840 


98831 
98828 
988. '5 
98822 
98819 
98816 

9.98813 
98810 


98804 


98798 
98795 


98789 
98786 


98777 
98774 
98771 


98765 
98762 
98759 
98756 

9.98753 
98750 


98743 
98740 
98737 
98734 
98731 


98725 

9.98722 
98719 
98715 
98712 
98709 
98706 
98703 
98700 


9.38368 
38418 


38519 
38570 
38620 


38721 
38771 
38821 


39021 
39071 
39121 
39170 
39220 
39270 
39319 


39418 
39467 
39517 
39566 
39615 


39713 
39762 
39811 


39958 
40006 
40055 
40103 
40152 
40200 
40249 
40297 

9.40346 


40442 
40490 
40538 
40586 
40634 


40730 
40778 


40873 
40921 
40968 
41016 
41063 
41111 
41158 
41205 
41252 
41300 


9.98690 


98675 
98671 
98668 
98665 
98662 


>.  38871   9.98659 


98652 
98649 


98643 


98633 
98030 

9.98627 
98623 


98614 


98607 
98604 


98597 
9.98594 


98588 
98584 
98581 
9S578 
98574 
98571 


98565 

9.98561 
98558 
98555 
98551 
98548 
98545 
98541 


98535 
98531 

9.-J8528 
98525 
98521 
98518 
98515 
98511 


98505 
98501 


98494 


Cosine   Sine 


Cosine         Sine         Cosine 
76° 


Sine 


140 


LOGARITHMIC  SINES  AND  COSINES 


/ 

15° 

16° 

17° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine  | 

0 

9.41300 

9.98494 

9.44034 

9.98284 

9.46594 

9.98060    60 

1 

41347 

98491 

44078 

98281 

46635 

98050 

59 

2 

41394 

98488 

44122 

98277 

46676 

98052 

58 

3 

41441 

98484 

44166 

98273 

46717 

98048 

57 

4 

41488 

98481 

44210 

98270 

46758 

9t?044 

56 

5 

41535 

98477 

44253 

98266 

46800 

98040 

55 

6 

41  582 

98474 

44297 

98262 

46841 

98036 

54 

7 

41028 

98471 

44341 

98259 

46882 

98032 

53 

8 

41675 

98467 

44385 

98255 

46923 

98029 

52 

9 

41722 

98464 

41428 

98251 

46964 

9b025 

51 

10 

9.41768 

9.98460 

9.44472 

9.98248 

9.47005 

9.98021 

50 

11 

41815 

98457 

44516 

9S244 

47045 

98017 

49 

12 

41861 

98453 

44559 

96240 

47086 

98013 

48 

13 

41908 

98450 

44602 

98237 

47127 

98009 

47 

14 

41954 

98447 

44646 

98233 

47168 

98005 

4(i 

15 

42001 

98  143 

44689 

98229 

47209 

98001 

45 

16 

42047 

98440 

44733 

98226 

47249 

97997 

44 

17 

42093 

98136 

44776 

98222 

47290 

97993 

43 

18 

42140 

ib'433 

44819 

98218 

47330 

97989 

42 

19 

42186 

98429 

44862 

98215 

47371 

97986 

41 

20 

9.-J2232 

9.98426 

9.44905 

9.98211 

9.47411 

9.97982 

40 

21 

42278 

98422 

44948 

98207 

47452 

97978 

39 

22 

42324 

98419 

44992 

98204 

47492 

97974 

38 

23 

42370 

98415 

45035 

98200 

47533 

97970 

37 

24 

42416 

98412 

45077 

98196 

47573 

97906 

36 

25 

42461 

98409 

45120 

98192 

47613 

97962 

35 

26 

42507 

98405 

45163 

98189 

47654 

97958 

34 

27 

42553 

98402 

45206 

98185 

47094 

97954 

33 

28 

42599 

98398 

45249 

98181 

47734 

97950 

3.' 

29 

42(344 

98395 

45292 

98177 

47774 

97946 

31 

30 

9.42690 

9.98391 

9.45334 

9.98174 

9.47814 

9.97942 

30 

31 

42735 

98388 

45377 

98170 

47854 

97938 

29 

32 

42781 

98384 

45419 

98166 

47894 

97934 

28 

33 

42826 

98381 

45462 

98162 

47934 

97930 

27 

34 

42872 

98377 

45504 

98159 

47974 

97926 

26 

35 

42917 

98373 

45547 

98155 

4f'014 

97922 

25 

36 

42962 

98370 

45589 

98151 

48054 

97918 

24 

37 

43008 

98366 

45632 

98147 

48094 

97914    23 

38 

43053 

98363 

45674 

98144 

48133 

97910  |  22 

39 

43098 

98:359 

45716 

98140 

48173 

97906    81 

40 

9.43143 

9.98356 

9.45758 

9.98136 

9.48213 

9.97902 

20 

41 

43188 

98352 

45801 

98132 

48252 

97898 

19 

42 

43528'* 

98349 

45843 

98129 

48292 

97894 

18 

43 

43278 

98345 

45885 

98125 

48332 

97890 

17 

44 

43323 

98342 

45927 

98121 

48371 

97886 

16 

45 

43367 

98338 

45969 

98117 

48411 

97882 

15 

46 

43412 

98334 

46011 

98113 

48450 

97878 

14 

47 

43457 

98331 

46053 

98110 

48490 

97874 

13 

48 

43502 

98327 

46095 

98106 

48529 

97870 

12 

49 

43546 

98324 

46136 

98102 

48568 

97866 

11 

50 

9.43591 

9.98320 

9.46178 

9.98098 

9.48607 

9.97861 

10 

51 

43635 

98317 

46220 

98094 

48647 

«   97857 

9 

52 

43680 

98313 

46262 

98090 

48686 

97853 

8 

53 

43724 

98309 

46303 

98087 

48725 

97849 

7 

54 

43769 

98306 

46345 

98083 

48764 

97845 

6 

55 

43813 

98302 

46386 

98079 

48803 

97841 

5 

56 

43857 

98299 

46428 

98075 

48842 

97837 

4 

57 

43901 

98295 

46469 

98071 

48881 

97833 

3 

58 

43946 

98291 

46511 

98067 

48920 

97829 

2 

59 

43990 

98288 

46552 

98063 

48959 

97825 

1 

60 

44034 

98284 

46594 

98060 

48998 

97821 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

74° 

78° 

72° 

LOGARITHMIC  SINES  AND  COSINES 


141 


18° 

19° 

20° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.48998 

9.97821 

9.51264 

9.97567 

9.53405 

9.97299 

60 

1 

49037 

97817 

51301 

97563 

53440 

97294 

59 

2 

490T6 

97812 

51338 

97558 

53475 

97289 

58 

3 

49115 

97808 

51374 

97554 

53509 

97285 

57 

4 

49153 

97804 

51411 

97550 

53544 

97280 

56 

5 

49192 

97800 

51447 

97545 

53578 

97276 

55 

6 

49231 

97796 

51484 

97541 

5S613 

97271 

54 

7 

49269 

97792 

51520 

97536 

53647 

97266 

53 

8 

49308 

97788 

51557 

97532 

53682 

97262 

52 

9 

49347 

97784 

51593 

97528 

53716 

97257 

51 

10 

9  49385 

9.97779 

9.51629 

9.97523 

9.53751 

9.97252 

50 

11 

49424 

97775 

51666 

97519 

53785 

97248 

49 

12 

494(52 

97771 

51702 

97515 

53819 

97243 

48 

13 

49500 

97767 

51738 

97510 

53854 

97238 

47 

14 

49539 

97763 

51774 

97506 

53888 

97234 

46 

15 

49577 

97759 

51811 

97501 

53922 

97229 

45 

16 

49615 

97754 

51847 

97497 

53957 

97224 

44 

17 

49654 

97750 

51883 

97492 

53991 

97220 

43 

18 

49692 

97746 

51919 

97488 

54025 

97215 

42 

19 

49730 

97742 

51955 

97484 

f  059 

97210 

41 

20 

9  497(58 

9.97738 

9.51991 

9.97479 

9.54093 

9.97206 

40 

21 

49806 

97734 

52027 

97475 

54127 

97201 

39' 

22 

49S14 

97729 

52063 

97470 

54161 

97196 

38 

23 

498  S-» 

97725 

52099 

97466 

54195 

97192 

37 

24 

499  ->0 

97721 

52135 

97461 

54229 

97187 

36 

25 

49958 

91717 

52171 

97457 

54263 

97182 

35 

26 

49996 

97713 

52207 

97453 

54297 

97178 

34 

27 

50034 

97708 

52242 

97448 

54331 

97173 

33 

28 

500;2 

97704 

52278 

97444 

54365 

97168 

32 

29 

50110 

97700 

52314 

97439 

54399 

97163 

31 

30 

9.50118 

9.97696 

9.52350 

9.97435 

9.54433 

9.97159 

30 

31 

50le>5 

97691 

52385 

97430 

54466 

97154 

29 

32 

50223 

97687 

52421 

97426 

54500 

97149 

28 

33 

50261 

97683 

52456 

97421 

54534 

97145 

27 

34 

50298 

97679 

52492 

97417 

54567 

97140 

26 

35 

50336 

97674 

52527 

97412 

54601 

97135 

25 

36 

50374 

97670 

52563 

97408 

54635 

97130 

24 

37 

50111 

97666 

52598 

97403 

54668 

97126 

23 

38 

50449 

97662 

52634 

97399 

54702 

97121 

22 

39 

50486 

97657 

52669 

97394 

64735 

97116 

21 

40 

9.50523 

9.97653 

9.52705 

9.97390 

9.54769 

9.97111 

20 

41 

50561 

97649 

52740 

97385 

54802 

97107 

19 

42 

50598 

97645 

52775 

97381 

54836 

97102 

18 

43 

50635 

97640 

52811 

97376 

54869 

97097 

17 

44 

50673 

97636 

52846 

97372 

54903 

97092 

16 

45 

50710 

97632 

52881 

97367 

54936 

97087 

15 

46 

50747 

97628 

52916 

97363 

54969 

97083 

14 

47 

50784 

97623 

52951 

97858 

55003 

97078 

13 

48 

50821 

97619 

52986 

97353 

55036 

97073 

12 

49 

50858 

97615 

53021 

97349 

55069 

97068 

11 

50 

9.50896 

9.97610 

9.53056 

9.97344 

9  55102 

9.97063 

10 

51 

509:53 

97606 

53092 

97340 

55136 

97059 

9 

52 

50970 

97602 

53126 

97335 

55169 

97054 

8 

53 

51007 

97597 

53161 

97331 

55202 

97049 

7 

54 

51043 

97593 

53196 

97326 

55235 

97044 

6 

55 

51080 

97589 

53231 

97322 

55268 

97039 

5 

56 

51117 

97584 

53266 

97317 

55301 

97035 

4 

57 

51154 

97580 

53301 

97312 

55334 

97030 

3 

58 

51  191 

97576 

53336 

97308 

55367 

97025 

2 

59 

51227 

97571 

53370 

97303 

55400 

97020 

1 

60 

51264 

97567 

53405 

97299 

55433 

97015 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

71° 

70° 

69° 

142 


LOGARITHMIC  SINES  AND  COSINES 


/ 

21°             22° 

23° 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.55433 

9.97015 

9.57358 

9.96717 

9.59188 

9.96403 

60 

1 

55466 

97010 

573«9 

96711 

59218 

96397 

59 

2 

55499 

97005 

57420 

96706 

59247 

96392 

58 

3 

55532 

97001 

57451 

96701 

59277 

96387 

57 

4 

55564 

96996 

57482 

96696 

59307 

96381 

56 

5 

55597 

96991 

57514 

96691 

59336 

96376 

55 

6 

55630 

96986 

57545 

96686 

59366 

96370 

54 

7 

556(33 

96981 

57576 

96681 

59396 

96365 

53 

8 

55695 

96976 

57607 

96676 

59425 

96360 

52 

9 

55728 

96971 

57638 

96670 

59455 

96354 

51 

10 

9.55761 

9.96966 

9.57669 

9.96665 

9.59484 

9.963-19 

50 

11 

55793 

96962 

57700 

96C60 

59514 

96343 

49 

12 

55826 

96957 

57731 

96655 

59543 

96338 

48 

13 

55858 

96952 

57762 

96650 

59573 

96333 

47 

14 

55891 

96947 

57793 

96645 

59602 

96327 

46 

15 

55923 

96942 

57824 

96640 

59632 

96322 

45 

16 

55956 

96937 

57855 

96634 

59661 

96316 

44 

17 

55988 

96932 

57885 

96629 

59690 

96311 

43 

18 

56021 

96927 

57916 

96624 

59720 

96305 

42 

19 

56053 

9G922 

57947 

96619 

59749 

96300 

41 

20 

I'.  56085 

9.96917 

9.57978 

9.96614 

9.59778 

9.96294 

40 

'  21 

561  18 

96912 

5800  •! 

96608 

59808 

96289 

39 

22 

56150 

90907 

58039 

96003 

59837 

96284 

38 

23 

56182 

96903 

58070 

96598 

59866 

96278 

37 

24 

56215 

96898 

58101 

96593 

59895 

96273 

36 

25 

56*247 

96893 

58131 

96588 

59924 

96267 

35 

26 

56279 

96888 

58162 

96582 

59954 

96262 

34 

27 

56311 

96883 

58192 

96577 

59963 

96256 

33 

28 

56343 

96878 

58223 

96572 

60012 

96251 

32 

29 

56375 

96873 

58253 

96567 

60041 

96245 

31 

30 

9.56108 

9.96868 

9.58284 

9.9G562 

9.60070 

9.96240 

30 

31 

56440 

96863 

58314 

96556 

60099 

96234 

29 

32 

56472 

96858 

58345 

96551 

60128 

96229 

28 

33 

56504 

96853 

58375 

96546 

60157 

962',3 

27 

34 

56536 

96848 

58406 

96541 

60186 

96218 

26 

a5 

56568 

96843 

58436 

06535 

60215 

96212 

25 

ae 

56599 

96838 

58467 

96530 

60244 

96207 

24 

37 

56631 

96833 

58497 

96525 

60273 

96201 

23 

38 

56663 

96828 

58527 

96520 

60302 

96196 

22 

39 

56695 

96823 

58557 

96514 

60331 

96190 

21 

40 

9.56727 

9.96818 

9.58588 

9.96509 

9.60359 

9.96185 

20 

41 

56759 

908  13 

58618 

96504 

60388 

96179 

19 

42 

56790 

96808 

58648 

96498 

60417 

96174 

18 

43 

56822 

96803 

58678 

96493 

60446 

96108 

17 

44 

56854 

96798 

58709 

96488 

60474 

96162 

16 

45 

56886 

96793 

58739 

96483 

60503 

V6157 

15 

46 

56917 

96788 

58769 

96477 

60532 

90151 

14 

47 

56949 

96783 

58799 

96472 

60561 

96146 

13 

48 

56980 

96778 

58829 

96467 

60589 

96140 

12 

49 

57012 

96772 

58859 

96461 

60618 

96135 

11 

50 

9.57044 

9.96767 

9.58889 

9.96456 

9.60646 

0.36189 

m 

51 

57075 

96762 

58919 

96451 

60075 

96123     9 

52 

57107 

96757 

58949 

96445 

60704 

96118     8 

53 

57138 

96752 

58979 

96440 

60732 

96112 

7 

54 

57169 

96747 

59009 

96435 

60761 

96107 

6 

55 

57201 

96742 

59039 

96129 

60789 

96101 

K 

56 

57232 

96737 

59069 

96424 

60818 

96095     d 

57 

57264 

96732 

59098 

9641'.) 

60846 

96090 

8 

58 

57295 

96727 

59128 

96413 

60875 

960S4 

2 

59 

57326 

96722 

59158 

96408 

60903 

96079 

1 

60 

57358 

96717 

59188 

96403 

60931 

96073 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine    Sine 

/ 

68° 

67° 

66° 

LOGARITHMIC  SINES  AND  COSINES 


143 


/ 

24° 

25° 

26° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

o 

9.60931 

9.96073 

9.62595 

9.95728 

9.64184 

9.95366 

60 

60960 

96067 

62622 

95722 

64210 

95360 

59 

2 

60988 

96062 

62649 

95716 

64236 

95354 

58 

3 

61016 

96056 

62676 

95710 

64262 

95348 

57 

4 

61045 

96050 

62703 

95704 

64288 

95341 

56 

5 

61073 

96045 

62730 

95698 

64313 

95335 

55 

6 

61101 

96039 

62757 

95692 

64339 

95329 

54 

7 

61129 

96034 

62784 

95686 

64365 

95323 

53 

8 

61158 

96028 

62811 

95680 

64391 

95317 

52 

9 

61186 

96022 

62838 

95674 

64417 

95310 

51 

10 

9.61214 

9.96017 

9.62865 

9.95668 

9.64442 

9.95304 

50 

11 

61242 

96011 

02892 

95663 

64468 

95298 

49 

12 

61270 

96005 

62918 

95657 

64494 

95292 

48 

13 

61*98 

96000 

62915 

95651 

64519 

95286 

47 

14 

61326 

95994 

62972 

95645 

64545 

95279 

46 

15 

61354 

95988 

62999 

95639" 

64571 

95273 

45 

16 

61382 

95982 

63026 

95633 

64596 

95267 

44 

17 

61411 

95977 

63052 

95627 

64622 

95261 

43 

18 

61438 

95971 

63079 

95621 

64G47 

95254 

42 

19 

61466 

95965 

63106 

95615 

64673 

95248 

41 

20 

9.61494 

9.95960 

9.63133 

9.95609 

9.64698 

9.95242 

40 

21 

61522 

95954 

63159 

95603 

64724 

95286 

39 

22 

61550 

95948 

63186 

95597 

647'49 

95229 

38 

23 

61578 

95942 

63213 

95591 

64775 

95223 

37 

24 

61606 

95937 

63239 

95585 

64800 

95217 

36 

25 

61634 

95931 

63266 

95579 

64826 

95211 

35 

26 

61662 

95925 

63292 

95573 

C4851 

95204 

34 

27 

61689 

95920 

63319 

95567 

64877 

95198 

33 

28 

61717 

95914 

63345 

95561 

64902 

95192 

32 

29 

61745 

95908 

63372 

95555 

64927 

95185 

31 

30 

9.61773 

9.95902 

9.63398 

9  95549 

9.64953 

9.95179 

30 

31 

61800 

95897 

63425 

95543 

64978 

95173 

29 

32 

61828 

95891 

63451 

95537 

65003 

95167    28 

33 

61856 

95885 

63478 

95531 

65029 

95160    27 

34 

61883 

95879 

C3504 

95525 

65054 

95154 

26 

35 

61911 

95873 

63531 

95519 

65079 

95148 

25 

«6 

61939 

95868 

63557 

95513 

65104 

95141 

24 

37 

61966 

95862 

63583 

95507 

65130 

95135 

23 

38 

61994 

95856 

63610 

95500 

65155 

95129 

22 

39 

62021 

95850 

63636 

95494 

65180 

95122 

21 

40 

9.62049 

9.95844 

9.63662 

9.95488 

9.65205 

9.95116 

20 

41 

62076 

95839 

63689 

95482 

65230 

95110 

19 

42 

62104 

95833 

63715 

95476 

65255 

95103 

18 

43 

62131 

95827 

63741 

95470 

65281 

95097 

17 

44 

62159 

95821 

63767 

95464 

65306 

95090 

16 

45 

62186 

95815 

63794 

95458 

65331 

95084 

15 

46 

62214 

95810 

63820 

95452 

65356 

95078 

14 

47 

62241 

95804 

63846 

95446 

65381 

95071 

13 

48 

62268 

95798 

63872 

95440 

65406 

95065 

12 

49 

62296 

95792 

63898 

95434 

65431 

95059 

.  11 

50 

9.62323 

9.95786 

9.63924 

9.95427 

9.65456 

9.95052 

10 

51 

62350 

95780 

63950 

95421 

65481 

95046 

9 

52 

62377 

95775 

63976 

95415 

65506 

95039 

8 

53 

62405 

95769 

64002 

95409 

65531 

95033 

7 

54 

62432 

95763 

64028 

95403 

65565 

95027 

6 

55 

62459 

95757 

64054 

95397 

65580 

95020 

5 

56 

62486 

95751 

64080 

95391 

65605 

95014 

4 

57 

62513 

95745 

64106 

95384 

65630 

95007 

8 

58 

62541 

95739 

64132 

95378 

65655 

95001 

2 

59 

62568 

95733 

64158 

95372 

65680 

94995 

1 

60 

62595 

95728 

64184 

95366 

65705 

94988 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

65° 

64° 

68° 

144 


LOGARITHMIC   SINES  AND   COSINES 


27° 

28° 

29° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.65705 

9.94988 

9.67161 

9.94593 

9.68557 

9.94182 

60 

1 

65729 

94982 

67186 

94587 

68580 

94175 

59 

2 

65754 

94975 

67208 

94580 

68603 

94168 

58 

3 

65779 

94969 

67232 

94573 

68625 

94161 

57 

4 

65804 

94962 

67256 

94567 

68648 

94154 

56 

5 

65828 

94956 

67280 

94560 

68671 

94147 

55 

6 

65853 

94949 

67303 

94553 

68694 

94140 

54 

7 

65878 

94943 

67327 

94546 

68716 

94133 

53 

8 

65902 

94936 

67350 

94540 

68739 

94126 

52 

9 

65927 

94930 

67374 

94533 

68762 

94119 

51 

10 

9.65952 

9.94923 

9.67398 

9.94526 

9.68784 

9.94112 

50 

11 

65976 

94917 

67421 

94ol9 

68807 

94105 

40 

12 

66001 

94911 

67445 

94513 

68829 

94098 

48 

II 

66025 

94904 

67468 

94506 

68852 

94090 

47 

14 

66050 

94898 

67492 

94499 

68875 

94083 

46 

15 

66075 

94891 

'67515 

94492 

68897 

94076    45 

16 

66099 

94885 

67539 

94485 

68920 

94069 

44 

17 

66124 

94878 

67562 

94479 

68942 

94062 

43 

18 

66148 

94871 

6758G 

94472 

68965 

94055 

42 

19 

66173 

94865 

67609 

94465 

68987 

94048 

41 

20 

9.66197 

9.94858 

9.67633 

9.94458 

9.69010 

9.94041 

40 

21 

66221 

94852 

67656 

94451 

69032 

94034 

39 

22 

60246 

94845 

67680 

94445 

69055 

94027 

38 

23 

66-J70 

94839 

67703 

94438 

69077 

94020 

37 

24 

66295 

94832 

67726 

94431 

69100 

94012 

36 

25 

66319 

94826 

67750 

94424 

69122 

94005 

35 

26 

66343 

94819 

67773 

94417 

69144 

93998 

34 

27 

66368 

94813 

67796 

94410 

69167 

93991 

33 

28 

66392 

94806 

67820 

94404 

69189 

93984 

82 

29 

66416 

94799 

67843 

94397 

69212 

93977 

31 

30 

9.66441 

9.94793 

9.67866 

9.94390 

9.69284 

9.93970 

80 

81 

66465 

94786 

67890 

94383 

69256 

93963 

29 

32 

66489 

94780 

67913 

94376 

69279 

93955 

28 

83 

66513 

94773 

67936 

94369 

69301 

93948 

27 

34 

66537 

94767 

67959 

94362 

69323 

08941 

26 

35 

66562 

94760 

67982 

94355 

69345 

93934 

25 

36 

66586 

94753 

68006 

94349 

69368 

93927 

24 

37 

66610 

94747 

C8029 

94342 

69390 

93920 

23 

38 

66634 

94740 

68052 

94335 

69412 

£3912 

22 

39 

66658 

94734 

6S075 

94328 

69434 

93905 

21 

40 

9.66682 

9.94727 

9.68098 

9.94321 

9.69456 

9.93898 

20 

41 

66706 

94720 

68121 

94314 

69479 

93891 

19 

42 

66731 

94714 

68144 

94307 

69501 

93884 

18 

43 

66755 

94707 

68167 

94300 

69523 

93876 

17 

44 

66779 

94700 

68190 

94293 

69545 

93869 

16 

45 

66803 

94694 

68213 

94286 

69567 

93862 

15 

46 

66827 

94687 

68237 

94279 

69589 

93855 

14 

47 

66851 

94680 

68260 

94273 

69611 

93847 

18 

48 

66875 

94674 

68283 

94266 

69638 

93840 

12 

49 

66899 

94667 

68305 

94259 

69655 

93833 

11 

50 

9.66922 

9.94660 

9.68328 

9.94252 

9.69677 

9.93826 

10 

51 

66946 

94654 

68351 

94245 

69699 

93819 

9 

52 

66970 

94647 

68374 

94238 

6.9721 

93811 

8 

53 

66994 

94640 

68397 

94231 

69743 

93804 

7 

54 

67018 

94634 

68420 

94224 

69765 

93797 

6 

55 

67042 

94627 

68443 

94217 

69787 

93789 

5 

56 

67066 

94620 

68466 

94210 

69809 

93782 

4 

57 

67090 

94614 

68489 

94203 

69831 

93775 

3 

58 

67113 

94607 

68512 

94196 

69853 

93768 

2 

59 

67137 

94600 

68534 

94189 

69875 

93760 

1 

60 

67161 

94593 

68557 

94182 

69897 

93753 

0 

t 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

, 

42° 

61° 

60° 

LOGARITHMIC  SINES  AND  COSINES 


145 


30° 

31° 

32° 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine    Cosine 

0 

9.69897 

9.93753 

9.71184 

9.93307 

9.72421   9 

92842 

60 

1 

69919 

93746 

71205 

93299 

72441 

92834 

59 

2 

69941 

93738 

71226 

93291 

72461 

9-2826 

58 

3 

69963 

93731 

71247 

93284 

72482 

92818 

57 

4 

69984 

93724 

71268 

93276 

72502 

9:2810 

56 

5 

70006 

93717 

71289 

93269 

72522 

92803 

55 

6 

70028 

93709 

71310 

93261 

72542 

92795 

54 

7 

70050 

93702 

71331 

93253 

72562 

9-2787 

53 

8 

70072 

93095 

71352 

93246 

72582 

92779 

52 

9 

70093 

93687 

71373 

93238 

7~>602 

92771 

51 

10 

9.70115 

9.93680 

9.71393 

9.932SO 

9  7-2622   9.92763 

50 

11 

70137 

93673 

71414 

932C3 

72643 

92155 

49 

12 

70159 

93665 

71435 

93215 

72663 

92747 

48 

13 

70180 

93658 

71456 

93-207 

72683 

9-2739 

47 

14 

70202 

93650 

71477 

93200 

72703 

92731 

46 

15 

70224 

93643 

71498 

93192 

72723 

92723 

45 

16 

70245 

93636 

71519 

93184 

72743 

92715 

44 

17 

70-267 

93628 

71539 

93177 

727G3 

92707 

43 

18 

70-288 

93621 

71560 

93169 

72783 

92699 

42 

19 

70310 

93614 

71581 

93161 

72803 

9-2691 

41 

20 

9.70332 

9.93606 

9.71602 

9.93154 

9.72823   9 

92683 

40 

21 

70353 

93599 

71622 

93146 

72843 

92675 

39 

22 

70375 

93591 

71643 

93138 

72863 

92667 

38 

23 

70396 

93584 

71664 

93131 

728S3 

92659 

37 

24 

70418 

93577 

71685 

93123 

72902 

92651 

36 

25 

70439 

93569 

71705 

93115 

72922 

92643 

35 

26 

70461 

93562 

71726 

93108 

72942 

92635 

34 

27 

70482 

93554 

71747 

93100 

72962 

92627 

33 

28 

70504 

93547 

71767 

93092 

72982 

92619 

32 

29 

70525 

93539 

71788 

93084 

73002 

92611 

31 

30 

9.70547 

9.93532 

9.71809 

9.93077 

9.73022   9 

92603 

30 

31 

70568 

93525 

71829 

93069 

73041 

92595 

29 

32 

70590 

93517 

71850 

93061 

73061 

92587 

28 

33 

70611 

93510 

71870 

93053 

73081 

92579 

27 

34 

70633 

93502 

71891 

93046 

73101 

92571 

26 

35 

70654 

93495 

71911 

93038 

73121 

9-2563 

25 

36 

•  70675 

93487 

71932 

93030 

73140 

92555 

24 

37 

70697 

93480 

71952 

93022 

73160 

9-2546 

23 

38 

70718 

93472 

71973 

9:  JO  14 

73180 

92538 

22 

39 

70739 

93465 

71994 

93007 

73200 

92530 

21 

40 

9.70761 

9.93457 

9.72014 

9.92999 

9.73219   9.92522 

20 

41 

70782 

93450 

72034 

9-2991 

73239 

92514 

19 

42 

70803 

93442 

72055 

92983 

73259 

92506 

18 

43 

70824 

93435 

72075 

92976 

73-278 

92498 

17 

44 

70846 

93427 

72096 

92968 

73298 

92490 

16 

45 

70867 

93420 

72116 

92960 

73318 

92482 

15 

46 

70888 

93412 

72137 

92952 

73337 

92473 

14 

47 

70909 

93405 

72157 

92944 

73357 

92465 

13 

48 

70931 

93397 

72177 

92936 

73377 

92457 

12 

49 

70952 

93390 

72198 

92929 

73396 

92449 

11 

50 

9.70973 

9.93382 

9.72218 

9.92921 

9.73416   9 

92441 

10 

51 

70994 

93375 

72238 

92913 

73435 

92433 

9 

52 

71015 

93367 

72259 

92905 

73455 

92425 

8 

53 

71036 

93360 

72279 

92897 

73474 

92416 

7 

54 

71058 

93352 

72299 

92889 

73494 

92408 

6 

55 

71079 

93344 

72320 

92881 

73513 

92400 

5 

56 

71100 

93337 

72340 

92874 

73533 

92392 

4 

57 

71121 

93329 

72360 

92866 

73552 

92384 

3 

58 

71142 

93322 

72381 

92858 

73572 

92376 

2 

59 

71163 

93314 

72401 

92850 

73591 

92367 

1 

60 

71184 

93307 

72421 

92842 

73611 

92359 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

69° 

58° 

57° 

146 


LOGARITHMIC  SINES  AND   COSINES 


88° 

84° 

85° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.73611 

9.92359 

9.74756 

9.91857 

9.75859 

9.5)1336 

60 

73630 

92351 

74775 

91849 

75877 

91328 

59 

2 

73650 

92343 

74794 

91840 

75895 

91319 

58 

3 

73669 

92335 

74812 

91832 

75913 

91310 

57 

4 

73C89 

92326 

74831 

91823 

75931 

91301 

56 

5 

73708 

92318 

74850 

91815 

75949 

91292 

55 

6 

73727 

92310 

74868 

91806 

75967 

91283 

54 

7 

73747 

92302 

74887 

91798 

75985 

91274 

53 

8 

73760 

92293 

74906 

91789 

76003 

91266 

52 

9 

73785 

92285 

74924 

91781 

76021 

91257 

51 

10 

9.73805 

9.92277 

9.74943 

9.91772 

9.76039 

9.91248 

50 

11 

73824 

92269 

74961 

91763 

76057 

91239 

49 

12 

73843 

92260 

74980 

91755 

70075 

91230 

48 

13 

73863 

92252 

74999 

91746 

76093 

91221 

47 

14 

73882 

92244 

75017 

91738 

76111 

91212 

46 

15 

73901 

92235 

75036 

91729 

76129 

91203 

45 

16 

73921 

92227 

75054 

91720 

76146 

91194 

44 

17 

73940 

92219 

75073 

91712 

76164 

91185 

43 

18 

73959 

92211 

75091 

91703 

76182 

91176 

42 

19 

73978 

92202 

75110 

91695 

76200 

91167 

41 

20 

9.73997 

9.92194 

9.75128 

9.  91  086 

9.76218 

9.91158 

40 

21 

74017 

92186 

75147 

91677 

76;.;36 

91149 

39 

22 

74036 

92177 

75165 

91669 

76253 

91141 

38 

23 

74055 

92169 

75184 

91660 

76271 

91132 

37 

24 

74074 

92161 

75'.'02 

91651 

76289 

91123 

36 

25 

74093 

92152 

75221 

91643 

76307 

91114 

IB 

26 

74113 

92144 

75239 

916:^4 

76324 

91105 

34 

27 

74132 

92136 

75258 

91625 

76342 

91096 

33 

28 

74151 

92127 

75276 

91617 

76360 

91087 

32 

29 

74170 

92119 

75294 

91608 

76378 

91078 

31 

30 

9.74189 

9.92111 

9.75313 

9.91599 

9.76395 

9.91069 

30 

31 

74208 

92102 

75331 

91591 

76413 

91060 

29 

32 

74227 

92094 

75350 

91582 

76431 

91051 

28 

33 

74246 

92086 

75368 

91573 

76448 

91042 

27 

34 

'<4265 

92077 

75386 

91565 

76466 

91033 

26 

35 

74284 

92069 

75405 

91556 

76484 

91023 

25 

36 

74303 

92060 

75423 

91547 

76501 

91014 

24 

3V 

74322 

92052 

75441 

91538 

76519 

91005 

23 

38 

74341 

92044 

75459 

91530 

76537 

90996 

22 

39 

74360 

92035 

75478 

91521 

76554 

90987 

21 

40 

9.74379 

9.92027 

9.75496 

9.91512 

9.76572 

9.90978 

20 

41 

74398 

92018 

75514 

91504 

76590 

90969 

19 

42 

74417 

92010 

75533 

91495 

76607 

90960 

18 

43 

74436 

92002 

75551 

91486 

76625 

90951 

17 

44 

74455 

91993 

75569 

91477 

76642 

90942 

16 

45 

74474 

91985 

75587 

91469 

76660 

90933 

15 

46 

74493 

91976 

75605 

91460 

76677 

90924 

14 

47 

74512 

91968 

75624 

91451 

76695 

90915 

13 

48 

74531 

91959 

75642 

91442 

76712 

90906 

12 

49 

74549 

91951 

75660 

91433 

76730 

90896 

11 

50 

9.74568 

9.91942 

9.75678 

9.91425 

9.76747 

9.90887 

10 

51 

74587 

91934 

75696 

91416 

76765 

90878 

9 

52 

74606 

91925 

75714 

91407 

76782 

90869 

8 

53 

74625 

91917 

75733 

91398 

76800 

90860 

f 

54 

74644 

91908 

75751 

91389 

76817 

90851 

6 

55 

74662 

91900 

75769 

91381 

76835 

90842 

5 

56 

74681 

91891 

75787 

91372 

76852 

90832 

4 

57 

74700 

91883 

75805 

91363 

76870 

90823 

3 

58 

74719 

91874 

75823 

91354 

76887 

90814 

2 

59 

74737 

91866 

75841 

91345 

76904 

90805 

1 

60 

74756 

91857 

75859 

91336 

76922 

90796 

0 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

f 

56° 

65° 

64° 

LOGARITHMIC  SINES  AND   COSINES 


147 


0 

36° 

37° 

3 

8° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.76922 

9.90796 

9.77946 

9.90235 

9.78934 

9.89653 

60 

1 

76939 

90787 

77963 

90225 

78950 

89643 

59 

2 

76957 

90777 

77980 

90216 

78967 

89633 

58 

3 

76974 

90768 

77997 

90206 

78983 

896-^4 

57 

4 

76991 

90759 

78013 

90197 

78999 

89614 

56 

5 

77009 

90750 

78030 

90187 

79015 

89tt)4 

55 

6 

77026 

90741 

78047 

90178 

79031 

89594 

54 

7 

77043 

90731 

78063 

90168 

79047 

89584 

53 

8 

77061 

90722 

78080 

90159 

79063 

89574 

52 

9 

77078 

90713 

78097 

90149 

79079 

89564 

51 

10 

9.77095 

9.90704 

9.78113 

9.90139 

9.79095 

9.89554 

50 

11 

77112 

90694 

78130 

90130 

79111 

89544 

49 

1ST 

77130 

90685 

78147 

90120 

79128 

89534 

48 

13 

77147 

90676 

78163 

90111 

79144 

89524 

47 

14 

77164 

90667 

78180 

90101 

79160 

89514 

46 

15 

77181 

90657 

78197 

90091 

79176 

89504 

45 

16 

77199 

90648 

78213 

90082 

79192 

89495 

44 

17 

77216 

90639 

78230 

90072 

79208 

89485 

43 

18 

77233 

90630 

78246 

90063 

79224 

89475 

42 

19 

77250 

90620 

78263 

90053 

79240 

89465 

41 

20 

9.77268 

9.90611 

9.7-280 

9.90043 

9.79256 

9.89455 

40 

21 

77285 

90602 

78296 

90084 

79272 

89445 

39 

22 

77302 

90592 

78313 

90024 

79288 

89435 

38 

*3 

77319 

90583 

78329 

90014 

79304 

89125 

37 

24 

77336 

90574 

78346 

90005 

79319 

89415 

36 

25 

77353 

90565 

78362 

89995 

79335 

89405 

35 

26 

77370 

90555 

78379 

89985 

79351 

89395 

34 

27 

77387 

90516 

78395 

89976 

79367 

89385 

33 

28 

77405 

90537 

78412 

89966 

79383 

89375 

32 

29 

77422 

905-27 

78428 

89956 

79399 

89364 

31 

30 

9.77439 

9.90518 

9.78445 

9.89947 

9.79415 

9.89354 

30 

31 

.77456 

90509 

78481 

89937 

79431 

89344 

29 

32 

77473 

90499 

78178 

89927 

79447 

89334 

28 

•  33 

77490 

90190 

78494 

80918 

79463 

89324 

27 

31 

77507 

90480 

78510 

89908 

79478 

89314 

26 

35 

77524 

90471 

78527 

89898 

79494 

89301 

25 

36 

77541 

90462 

78543 

89888 

79510 

89294 

24 

37 

77558 

90452 

78560 

89879 

79526 

89284 

23 

38 

77575 

90443 

78576 

89869 

79542 

89274 

22 

39 

77592 

90434 

78592 

89859 

79558 

89264 

21 

40 

9.77609 

9.90424 

9.78609 

9.89849 

9.79573 

9.89254 

20 

41 

77626 

90415 

78625 

89840 

79589 

89244 

19 

42 

77643 

90405 

78642 

89830 

79605 

89233 

18 

43 

77660 

90396 

78658 

89820 

78621 

89223 

17 

44 

77677 

90386 

78674 

89810 

79636 

89213 

16 

45 

77694 

90377 

78691 

89801 

79652 

89203 

15 

46 

77711 

9036S 

78707 

89791 

79668 

89193 

14 

47 

77728 

90353 

78723 

89781 

79684 

89183 

13 

48 

77744 

90349 

78739 

89771 

79699 

89173 

12 

49 

77761 

90339 

78756 

89761 

79715 

89162 

11 

50 

9.77778 

9.90330 

9.78772 

9.89752 

9.79731 

9.89152 

10 

51 

77795 

90320 

78788 

89742 

79746 

89142 

9 

52 

77812 

90311 

78805 

89732 

79762 

89132 

8 

53 

77829 

90301 

78821 

89722 

79778 

89122 

7 

54 

77846 

90292 

78837 

89712 

79793 

89112 

6 

55 

77862 

90282 

78853 

89702 

79809 

89101 

5 

56 

77879 

90273 

78869 

89693 

79825 

89091 

4 

57 

77896 

90263 

78886 

89683 

79840 

89081 

3 

58 

77913 

90254 

78902 

89673 

79856 

89071 

2 

59 

77930 

90244 

78918 

89663 

79872 

89060 

1 

60 

77946 

90235 

78934 

89653 

79887 

89050 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

68° 

62° 

t 

>1° 

148 


LOGARITHMIC   SINES   AND   COSINES 


39° 

40° 

41° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.79887 

9.89050 

9.80807 

9.88425 

9.81694 

9.87778 

60 

1 

79903 

89040 

80822 

88415 

81709 

87767 

59 

2 

79918 

89030 

80837 

88404 

81723 

87756 

58 

3 

79934 

890*) 

80852 

88394 

81738 

87745 

t^ 

4 

79950 

89009 

80867 

88383 

81752 

87734 

56 

5 

79965 

88999 

80882 

88372 

81767 

87723 

55 

6 

79981 

88989 

80897 

88362 

81781 

87712 

54 

7 

79996 

88978 

80912 

88351 

81796 

87701 

53 

8 

80012 

88968 

80927 

88340 

81810 

87690  •  52 

9 

80027 

88958 

80942 

88330 

81825 

87679    51 

10 

9.80043 

9.88948 

9.80957 

9.88319 

9.81839 

9.87668  '  50 

11 

80058 

8H937 

80972 

88308 

81854 

87657  '  49 

12 

80074 

88927 

80987 

88298 

81868 

87646    48 

13 

80089 

88917 

81002 

88287 

81882 

87635    47 

14 

80105 

88906 

81017 

88276 

81897 

87624  j  46 

15 

80120 

88896 

81032 

88266 

81911 

87613  I  45 

16 

80136 

88886 

81047 

88255 

81926 

87601 

44 

17 

80151 

88875 

81061 

88244 

81940 

87590 

43 

18 

80166 

88865 

81076 

88234 

81955 

87579 

42 

19 

80182 

88855 

81091 

88223 

81969 

87568 

41 

20 

9.80197 

9.88844 

9.81106 

9.88212 

9.81983 

9.87557 

40 

21 

80213 

88834 

81121 

88201 

81998 

87546 

39 

22 

80228 

88824 

81136 

88191 

82012 

87535 

38 

23 

80244 

88813 

81151 

88180 

82026 

87524 

37 

24 

80259 

88803 

81166 

88169 

82041 

87513 

36 

25 

80274 

88793 

81180 

88158 

82055 

87501    85 

26 

80290 

88782 

81195 

88148 

82069 

87490    34 

27 

80305 

88772 

81210 

•88137 

82084 

87479    33 

28 

80320 

88761 

81225 

88126 

82098 

87468    32 

28 

80336 

88751 

81240 

88115 

82112 

87457    31 

30 

9.80351 

9.88741 

9.81254 

9.88105 

9.82126 

9.87446 

30 

31 

80366 

88730 

81269 

88094 

82141 

87434 

29 

32 

80382 

88720 

81284 

88083 

82155 

87423 

28 

33 

80397 

88709 

81299 

88072 

82169 

87412 

27 

34 

80412 

88699 

81314 

88061 

82184 

87401 

26 

35 

80428 

88688 

81328 

88051 

82198 

87390 

25 

36 

80443 

88678 

81343 

88040 

82212 

87378 

24 

37 

80458 

88668 

81358 

88029 

82226 

87367 

23 

38 

80473 

88657 

81372 

88018 

82240 

87356 

22 

39 

80489 

88647 

81387 

88007 

82255 

87345 

21 

40 

9.80504 

9.88636 

9.81402 

9.87996 

9.82269 

9.87334 

20 

41 

80519 

88626 

81417 

87985 

82283 

87322 

19 

42 

80534 

88615 

81431 

87975 

82297 

87311 

18 

43 

80550 

88605 

81446 

87964 

82311 

87300 

17 

44 

80565 

88594 

81461 

87953 

82326 

87288 

16 

45 

80580 

88584 

81475 

87942 

821340 

87277 

15 

46 

80595 

8*573 

81490 

87931 

82354 

87266 

14 

47 

80610 

88563 

81505 

87920 

82368 

87255 

18 

48 

80625 

88552 

81519 

87909 

82382 

87243 

12 

49 

80641 

88542 

81534 

87898 

82396 

87232 

11 

50 

9.80656 

9.88531 

9.81549 

9.87887 

9.82410 

9.87S21 

10 

51 

80671 

88521 

81563 

87877 

82424 

87209 

9 

52 

80686 

88510 

81578 

87866 

82439 

87198 

8 

53 

80701 

88499 

81592 

87855 

824f3 

87187 

7 

54 

80716 

88489 

81607 

87844 

82467 

87175 

6 

55 

80731 

88478 

81622 

87833 

82481 

87164 

5 

56 

80746 

88468 

81686 

87822 

82495 

87153 

4 

57 

80762 

88457 

81651 

87811 

82509 

87141 

3 

58 

80777 

88447 

81665 

87800 

82523 

87130 

2 

59 

80792 

88436 

81680 

87789 

82537 

87119 

1 

60 

80807 

88425 

81694 

87778 

82551 

87107 

0 

~ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

, 

50° 

49° 

48° 

LOGARITHMIC  SINES  AND  COSINES 


149 


/ 

42° 

48° 

44° 

] 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

o 

9.82551 

9.87107 

9.88378 

9.86413 

9.84177 

9.85693 

60 

82565 

87096 

83392 

86401 

84190 

85681 

59 

2 

82579 

87085 

83405 

8G889 

84203 

85669 

58 

3 

82593 

87073 

83419 

86377 

84216 

85657 

57 

4 

82607 

87062 

83432 

86366 

84229 

85645 

56 

5 

82621 

87050 

83446 

86354 

84242 

85632 

55 

6 

82635 

87039 

83459 

86342 

84255 

85620 

54 

7 

82649 

87028 

83473 

86330 

84269 

85608 

53 

8 

82663 

87016 

83486 

86318 

84282 

85596 

52 

9 

82677 

87005 

83500 

86306 

84295 

85583 

51 

10 

9.82691 

9.86993 

9.83513 

9.86295 

9.84308 

9.85571 

50 

11 

82705 

86982 

83527 

86283 

84321 

85559 

49 

12 

82719 

86970 

83540 

86271 

84334 

85547 

48 

13 

82733 

86959 

83554 

86259 

84347 

85534 

47 

14 

82747 

86947 

83567 

86247 

84360 

85522 

46 

15 

82rei 

86936 

83581 

86235 

84373 

85510 

45 

16 

82775 

86924 

83594 

80888 

84385 

85497 

44 

17 

82788 

86913 

83608 

86211 

84398 

85485 

43 

18 

82802 

86902 

83621 

86200 

84411 

85473 

42 

19 

82816 

86890 

83634 

86188 

84424 

85460 

41 

20 

9.82830 

9.86879 

9.83648 

9.86176 

9.84437 

9.85448 

40 

21 

82844 

86867 

83661 

86164 

84450 

85436 

39 

22 

82858 

86855 

83674 

86152 

84463 

85423 

38 

23 

82872 

86844 

83688 

86140 

84476 

85411 

37 

24 

82885 

86832 

83701 

86128 

84489 

85399 

36 

25 

82899 

86821 

83715 

86116 

84502 

85386 

35 

26 

82913 

86809 

83728 

86104 

84515 

85874 

34 

27 

82927 

86798 

83741 

86092 

84528 

65361 

33 

28 

82-941 

86786 

83755 

86080 

84540 

85349 

32 

29 

82955 

86775 

83768 

86068 

84553 

85337 

31 

30 

9.82968 

9.86763 

9.83781 

9.86056 

9.84566 

9.85324 

30 

31 

82982 

86752 

83795 

86044 

84579 

85312 

29 

32 

82996 

86740 

83808 

86032 

84592 

85299 

28 

33 

83010 

86728 

83821 

86020 

84605 

85287 

27 

34 

83023 

86717 

83834 

86008 

84618 

85274 

26 

35 

88037 

86705 

83848 

85996 

84630 

85262 

25 

36 

83051 

86694 

83861 

85984 

84643 

85250 

24 

37 

83065 

86682 

83874 

85972 

84656 

85237 

23 

38 

83078 

86670 

83887 

859CO 

84669 

85225 

22 

39 

83092 

86659 

83901 

85948 

84682 

85212 

21 

40 

9.83106 

9.86647 

9.83914 

9.85936 

9.84694 

9.85200 

20 

41 

83120 

86635 

83927 

85924 

84707 

85187 

19 

42 

83133 

86624 

83940 

85912 

84720 

85175 

18 

43 

83147 

86612 

83954 

85900 

84733 

85162 

17 

44 

83161 

86600 

83967 

85888 

84745 

85150 

16 

45 

83174 

86589 

83980 

85876 

84758 

85137 

15 

46 

83  188 

86577 

83993 

85864 

84771 

85125 

14 

47 

83202 

86565 

84006 

85851 

81784 

85112 

13 

48 

83215 

86554 

84020 

85839 

84796 

85100 

12 

49 

83229 

86542 

840.33 

85827 

84809 

85087 

11 

50 

9.83242 

9.86530 

9.84046 

9.85815 

9.84822 

9.85074 

10 

51 

83256 

86518 

84059 

85803 

84835 

85062 

9 

52 

83270 

86507 

84072 

85791 

84847 

85049 

8 

53 

83283 

86495 

84085 

85779 

84860 

85037 

7 

54 

83297 

86483 

84098 

85766 

84873 

86024 

6 

55 

83310 

86472 

84112 

85754 

84885 

85012 

5 

56 

83324 

86460 

84125 

85742 

84898 

84999 

4 

57 

83338 

86448 

84138 

85730 

84911 

84986 

3 

58 

83351 

86436 

84151 

85718 

84923 

84974 

2 

59 

83365 

86425 

84164 

85706 

84936 

84961 

1 

60 

83378 

86413 

84177 

85693 

84949 

84949 

0 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

| 

470 

46° 

46° 

150 


LOG.   TANGENTS  AND   COTANGENTS 


/ 

0° 

1° 

20 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

—  oo 

OD 

8.21192 

11.75808 

8.54308 

11.45692 

60 

1 

6.46373 

13.53627 

24910 

75090 

54669 

45331 

59 

2 

76476 

23524 

25616 

74384 

55027 

44973 

58 

3 

94085 

05915 

26312 

73688 

55382 

44618 

57 

4 

7.06579 

12.93421 

26996 

73004 

55734 

44266 

56 

5 

16270 

83730 

27669 

72331 

56083 

43917 

55 

6 

24188 

75812 

28332 

71668 

56429 

43571 

54 

7 

30882 

69118 

28986 

71014 

56773 

43227 

53 

8 

36682 

63318 

29629 

70371 

57114 

42S86 

52 

9 

41797 

58203 

30263 

69737 

57452 

42548 

51 

10 

7.46373 

12.53627 

8.30888 

11.69112 

8.57788 

11.42212 

50 

11 

50512 

49488 

31505 

68495 

58121 

41879 

49 

12 

54291 

45709 

32112 

67888 

58451 

41549 

48 

13 

57767 

42233 

32711 

67289 

58779 

41221 

47 

14 

60986 

39014 

33302 

66698 

59105 

40895 

46 

15 

6398-2 

36018 

33886 

66114 

59428 

40572 

45 

16 

66785 

33215 

34461 

65539 

59749 

40251 

44 

17 

(19418 

30582 

35029 

64971 

60068 

39932 

43 

18 

71900 

28100 

35590 

64410 

60384 

39616 

42 

19 

74248 

25752 

36143 

63857 

60698 

89302 

41 

20 

7  76476 

12.23524 

8.36689 

11.63311 

8.61009 

11.38991 

40 

21 

78595 

21405 

37229 

62771 

61319 

38681 

39 

22 

80615 

19385 

37762 

62238 

61626 

38374 

38 

23 

82546 

17454 

38289 

61711 

61081 

38069 

37 

24 

S4394 

15606 

38809 

61191 

62234 

37766 

36 

25 

86167 

13833 

39323 

.   60677 

62535 

37465  i  35 

26 

87871 

12129 

39832 

60168 

62834 

371  C6    34 

27 

89510 

10490 

40334 

59666 

63131 

36869 

33 

28 

91089 

08911 

40880 

59170 

63426 

36574 

32 

29 

92613 

07387 

41321 

58679 

63718 

36282 

31 

30 

7.94086 

12.05914 

8.41807 

11.58193 

8.64009 

11.35991 

30 

31 

95510 

04490 

42287 

57713 

64298 

35702 

29 

32 

96889 

03111 

42762 

57238 

64585 

35415 

28 

33 

98225 

01775 

43232 

56768 

64870 

35130 

27 

34 

99522 

00478 

43696 

56304 

05154 

34846 

26 

35 

8.00781 

11.99219 

44156 

55844 

65435 

34565 

25 

36 

02004 

97996 

44611 

55389 

65715 

34285 

24 

37 

03194 

96S06 

45061 

54939 

65993 

34007 

23 

38 

04353 

95647 

45507 

54493 

66269 

33731 

22 

39 

05481 

94519 

45948 

54052 

66543 

33457 

21 

40 

8.06581 

11.93419 

8.46885 

11.53615 

8.66816 

11.23184 

20 

41 

07653 

92347 

46817 

53183 

67087 

32913 

19 

42 

08700 

91300 

47245 

52755 

67356 

32644 

18 

43 

09722 

90278 

47669 

52331 

67624 

32376 

17 

44 

10720 

89280 

48089 

51911 

67890 

32110 

16 

45 

11696 

88304 

48505 

51495 

68154 

31846 

15 

46 

12651 

87349 

48917 

51083 

68417 

31583 

14 

47 

13585 

86415 

493:25 

50675 

68678 

31322 

13 

48 

14500 

85500 

49729 

50271 

68938 

31062 

12 

49 

15395 

84605 

50130 

49870 

69196 

30804 

11 

50 

8.16273 

11.83727 

8.50527 

11  .49473 

8.69453 

11.30547 

10 

51 

17133 

82867 

50920 

49080 

69708 

30292 

9 

52 

17976 

82024 

51310 

48690 

69962 

30038 

8 

53 

18804 

81196 

51696 

48304 

70214 

29786 

7 

54 

19616 

80384 

52079 

47921 

70465 

29535 

6 

55 

20413 

79587 

52459 

47541 

70714 

29286 

5 

56 

21195 

78805 

52835 

47165 

70962 

29038 

4 

57 

21964 

78036 

53208 

46792 

71208 

28792 

3 

58 

22720 

77280 

53578 

46422 

71453 

28547 

2 

59 

23462 

76538 

53945 

46055 

71697 

28303 

1 

60 

24192 

75808 

54308 

45692 

71940 

28060 

0 

/ 

Cotau 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

89° 

88° 

87° 

LOG.   TANGENTS  AND   COTANGENTS 


151 


/ 

8° 

4° 

5° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

8.71940 

11.28060 

8.84464 

11.15536 

8.94195 

11.05805 

60 

I 

72181 

27819 

84646 

15354 

94340 

05660 

59 

2 

72420 

27580 

84826 

15174 

94485 

05515 

58 

3 

72659 

27341 

85006 

14994 

94630 

05370 

57 

4 

72896 

27104 

85185 

14815 

94773 

05227 

56 

5 

73132 

26868 

85363 

14637 

94917 

05083 

55 

6 

73366 

26634 

85540 

14460 

95060 

04940 

54 

7 

73600 

26400 

85717 

14283 

95202 

04798 

53 

8 

73333 

2616S 

85893 

14107 

95344 

04656 

52 

9 

74063 

25937 

86069 

13931 

95486 

04514 

51 

10 

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9.02162 

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31 

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10 

9.43308 

10.56692 

9.4G224 

10.53776 

9.48984 

10.51016 

50 

11 

43358 

56642 

46271 

53729 

49029 

50971 

49 

12 

43408 

56592 

46319 

53681 

49073 

50927 

48 

18 

43458 

56542 

46366 

53634 

49118 

50882 

47 

14 

43508 

56492 

46413 

53587 

49163 

50837 

46 

15 

43558 

56442 

46460 

53540 

49207 

50793 

45 

16 

43607 

56393 

46507 

53493 

49252 

50748 

44 

17 

43657 

56343 

46554 

5344G 

49-^96 

50704 

43 

18 

43707 

56293 

46601 

53399 

49341 

50659 

42 

19 

43756 

56244 

46648 

53352 

49385 

50615 

41 

20 

9  43806 

10.56194 

9.46694 

10.53306 

9.49430 

10.50570 

40 

21 

43855 

56145 

46741 

53259 

49474 

50526 

39 

22 

43905 

56095 

46788 

53212 

49519 

50481 

38 

23 

43954 

56046 

46835 

53165 

49563 

50437 

37 

24 

44004 

55996 

46881 

53119 

49607 

50393 

S6 

25 

44053 

55947 

46928 

53072 

49652 

50348 

35 

26 

44102 

55898 

46975 

53025 

49696 

50304 

34 

27 

44151 

55849 

47021 

52979 

49740 

50260 

33 

28 

44201 

55799 

47068 

52932 

49784 

50216 

3.' 

29 

44250 

55750 

47114 

52886 

49828 

50172 

31 

30 

9.44299 

10.55701 

9.47160 

10.52840 

9.49872 

10.50128 

30 

31 

41348 

55652 

47207 

52793 

49916 

50084 

29 

32 

44397 

55603 

47253 

52747 

49960 

50010 

28 

83 

44446 

55554 

47299 

52701 

50004 

49996 

27 

34 

44495 

55505 

47346 

52654 

50048 

49952 

26 

35 

44544 

55456 

47392 

52608 

50092 

49908 

25 

36 

44592 

55108 

47438 

52562 

50136 

49864 

24 

37 

44641 

55359 

47484 

52516 

50180 

49820 

23 

38 

44690 

55310 

47E30 

52470 

50223 

49777 

22 

39 

44738 

55262 

47576 

52424 

50267 

49733 

21 

40 

9.44787 

10.55213 

9.47622 

10.52378 

9.50311 

10.49689 

20 

41 

44836 

55164 

47668 

52332 

50355 

49645 

19 

42 

44884 

55116 

47714 

52286 

50398 

4960-3 

18 

43 

44933 

55067 

47760 

52240 

50442 

49558 

17 

44 

44981 

5.5019 

47806 

52194 

50485 

49515 

16 

45 

45029 

54971 

4785:? 

52148 

50529 

49471 

15 

46 

45078 

54922 

47897 

52103 

50572 

49428 

14 

47 

45126 

54874 

47943 

52057 

50616 

493C4 

13 

48 

45174 

5482(5 

47989 

52011 

50659 

49341 

12 

49 

45222 

54778 

48035 

51965 

50703 

49297 

11 

50 

9.45271 

10.54729 

9.48080 

10.51920 

9.50746 

10.49254 

JO 

51 

45319 

54681 

48126 

51874 

50789 

49211 

9 

52 

45367 

54633 

48171 

51829 

50833 

491G7 

8 

53 

45415 

54585 

48217 

51783 

50876 

49124 

7 

54 

45463 

54537 

48262 

51738 

50919 

49081 

6 

55 

45511 

54489 

48307 

51693 

50962 

49088 

5 

56 

45559 

54441 

48353 

51647 

51005 

48995 

4 

57 

45606 

54394 

48398 

51602 

51048 

48952 

3 

58 

45654 

54346 

48443 

51557 

51092 

48908 

2 

59 

45702 

54298 

48489 

51511 

51135 

48H65 

60 

45750 

54250 

48534 

51466 

51178 

48822 

0 

/ 

Co  tan 

Tan 

Cotan 

Tan 

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74° 

73° 

72° 

156 


LOG.   TANGENTS  AND  COTANGENTS 


/ 

18° 

19° 

20° 

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Tan 

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Tan 

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Tan 

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0 

9.51178 

10.48822 

9.53697 

10.46303 

9.56107 

10.43893 

60 

1 

51221 

48779 

53738 

46262 

56146 

43854 

59 

2 

51264 

48736 

53779 

46221 

56185 

438.5 

58 

3 

51306 

48694 

53820 

46180 

56-J24 

43776 

57 

4 

51349 

48651 

53861 

46139 

56264 

43730 

56 

5 

51392 

48608 

53902 

46098 

56303 

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55 

6 

51435 

48565 

53943 

46057 

56342 

43658 

54 

7 

51478 

48522 

53984 

46016 

56381 

43619 

53 

8 

51520 

48480 

54025 

45975 

56420 

43580 

52 

9 

51563 

4S437 

54065 

45935 

56459 

43541 

51 

10 

9.51606 

10.48394 

9.54106 

10.45894 

9.56498 

10.43502 

50 

11 

51648 

48352 

54147 

4S&53 

56537 

43463 

49 

12 

51691 

48309 

54187 

45813 

56576 

43424 

48 

13 

51734 

48266 

54228 

45772 

56815 

43385 

47 

14 

51770 

48224 

54269 

45731 

56654 

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46 

15 

51819 

48181 

54309 

45691 

56693 

43307 

45 

16 

51861 

48139 

54350 

45650 

56732 

43268 

44 

17 

51903 

48097 

54390 

45610 

56771 

4:'2J9 

43 

18 

51946 

48054 

54431 

45569 

56810 

43190 

42 

19 

51988 

48012 

54471 

45529 

56849 

43151 

41 

20 

9.52031 

10.47969 

9.54512 

10.45488 

9.56887 

10.43118 

40 

21 

52073 

47927 

54552 

45448 

56026 

43074 

39 

22 

52115 

47885 

54593 

45407 

5(5965 

43035 

38 

23 

52157 

47843 

54633 

45367 

57004 

42996 

37 

24 

52200 

47800 

54673 

45327 

57042 

42958 

30 

25 

52244 

47758 

54714 

45286 

57081 

42919 

35 

26 

52284 

47716 

54754 

45246 

57120 

42880 

34 

27 

52326 

47674 

54794 

45206 

57158 

42842 

33 

28 

52368 

47632 

54835 

45165 

57197 

42803 

82 

29 

52410 

47590 

54875 

45125 

57235 

42765 

31 

30 

9.52452 

10.47548 

9.54915 

10.45085 

9.57274 

10.42726 

30 

31 

52494 

47506 

54955 

45045 

57312 

42688 

29 

32 

52536 

47464 

54995 

45005 

57851 

42649 

28 

33 

52578 

47422 

55035 

44965 

57389 

42611 

27 

34 

52620 

47380 

55075 

44925 

57428 

42572 

26 

;-;.-. 

52661 

47339 

55115 

44885 

57466 

42534 

25 

36 

52703 

47'.'97 

55155 

44845 

57504" 

42496 

24 

37 

52745 

47255 

55195 

44805 

57543 

42457 

23 

38 

52787 

47213 

55235 

44765 

57581 

42419 

22 

39 

52829 

47171 

55275 

44725 

576)9 

42381 

21 

40 

9.52870 

10.47130 

9.55315 

10.44685 

9.57658 

10.42342 

20 

41 

52912 

47088 

65855 

44645 

57696 

42304 

19 

42 

52953 

47047 

55395 

44605 

57734 

42266 

18 

43 

52995 

47005 

55434 

44566 

57772 

42-^28 

17 

44 

53037 

46963 

55474 

445-26 

57810 

42190 

16 

45 

53078 

4G922 

55514 

44486 

57849 

42151 

15 

46 

53120 

46880 

55554 

44446 

57887 

42113 

14 

47 

53161 

46839 

55593 

44407 

57925 

42075 

13 

48 

53202 

46798 

55633 

44367 

579C3 

42037 

12 

49 

53244 

46756 

55673 

44327 

58001 

41999 

11 

50 

9.53285 

10.46715 

9.55712 

10.44288 

9.58039 

10.41961 

10 

51 

53327 

46673 

55752 

44248 

58077 

41923 

9 

52 

53368 

46632 

55791 

44209 

58115 

41885 

8 

53 

53409 

46591 

55831 

44169 

58153 

41847 

7 

54 

53450 

46550 

55870 

44130 

58191 

41809 

6 

55 

53492 

46508 

55910 

44090 

58229 

41771 

5 

56 

53533 

46467 

55949 

44051 

58267 

41733 

4 

57 

53574 

46426 

55989 

44011 

58304 

41696 

3 

58 

53615 

46385 

56028 

43972 

58342 

41658 

2 

59 

53656 

46344 

56067 

43933 

58380 

41620 

1 

60 

53697 

46303 

56107 

43893 

58418 

41582 

0 

i 

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Tan 

Cotan 

Tan 

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Tan 

/ 

71° 

70° 

69° 

LOG.  TANGENTS  AND  COTANGENTS 


157 


0 

21° 

22° 

23° 

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Tan 

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Tan 

Cotau 

Tan 

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o 

9.58418 

10.41582 

9.60641 

10.39359 

9.62785 

10.37215 

60 

1 

58455 

41545 

60677 

39323 

62820 

37180 

59 

2 

58493 

41507 

60714 

39286 

62855 

37145 

58 

3 

58531 

41469 

60750 

39250 

62890 

37110 

57 

4 

58569 

41431 

60786 

39214 

62926 

37074 

56 

5 

58606 

41394 

60823 

39177 

62961 

37039 

55 

6 

58644 

41356 

60859 

39141 

62996 

37004 

54 

58681 

41319 

60895 

39105 

63031 

36969 

53 

8 

58719 

41281 

60931 

39069 

63066 

36934 

52 

9 

58757 

41243 

60967 

39033 

63101 

36899 

51 

10 

9.58794 

10.41206 

9.61004 

10.38996 

9.63135 

10.36865 

50 

11 

58832 

41168 

61040 

38960 

63170 

36830 

49 

12 

58869 

41131 

61076 

38924 

63205 

36795 

48 

13 

58907 

41093 

61112 

38888 

63240 

36760 

47 

14 

58944 

41056 

61148 

38852 

63275 

36725 

46 

15 

58981 

41019 

61184 

•   38816 

63310 

3U690 

45 

16 

59019 

40981 

61220 

38780 

68:345 

36655 

44 

17 

59056 

40J4J 

61256 

38744 

63379 

36621 

43 

18 

59094 

40906 

61292 

38708 

63414 

36586 

42 

19 

59131 

40869 

61328 

38672 

63449 

36551 

41 

20 

9.59168 

10.40882 

9.61364 

10.38636 

9.63484 

10.36516 

40 

21 

59205 

40795 

61400 

38600 

63519 

36481 

39 

22 

59243 

40757 

61430 

8856* 

63553 

36447 

38 

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59280 

40720 

61472 

38528 

63588 

36412 

37 

24 

59317 

40683 

61508 

3^492 

63623 

36377 

36 

25 

59854 

40646 

61544 

38456 

63657 

36343 

35 

26 

59391 

40609 

61579 

38421 

03692 

36308 

34 

27 

59429 

40571 

61615 

38385 

03726 

36274 

33 

28 

59466 

4(1534 

61651 

38349 

63761 

36219 

32 

29 

59503 

40497 

61687 

38313 

63796 

36204 

31 

30 

9.59540 

10.40460 

9.61722 

10.38278 

9.63830 

10.36170 

30 

31 

59577 

40423 

C.1758 

3S242 

63865 

36135 

29 

32 

59614 

40386 

61794 

38206 

63899 

36101 

28 

33 

59651 

40349 

61830 

38170 

63934 

86066 

27 

34 

59688 

40312 

61865 

38135 

63968 

36032 

26 

35 

59725 

40275 

61901 

38099 

64003 

35997 

25 

36 

59762 

40238 

61936 

38064 

64037 

35963 

24 

37 

59799 

40201 

61972 

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64072 

35928 

23 

38 

59835 

40165 

62008 

37992 

64106 

35894 

22 

39 

59872 

40128 

62043 

37-957 

64140 

35860 

21 

40 

9.59909 

10.40091 

9.62079 

10.37921 

9.64175 

10.35825 

20 

41 

59946 

40054 

62114 

37886 

64209 

35791 

19 

42 

59983 

40017 

62150 

37850 

64243 

35757 

18 

43 

60019 

39981 

62185 

37815 

64278 

35722 

17 

44 

60056 

39944 

62221 

37779 

6*312 

35688 

16 

45 

60093 

39907 

62256 

37744 

64346 

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15 

46 

(50130 

39870 

62292 

37708 

64381 

35619 

14 

47 

60166 

39834 

62327 

37673 

64415 

35585 

13 

48 

60203 

39797 

62362 

37638 

64449 

35551 

12 

49 

60240 

39760 

62398 

37602 

64483 

35517 

11 

50 

9.60276 

10.39724 

9.62433 

10.37567 

9.64517 

10.35483 

10 

51 

60313 

39687 

62468 

37532 

64552 

•   35448 

9 

52 

60349 

39651 

62504 

37496 

64586 

35414 

8 

53 

60386 

39614 

62539 

37461 

64620 

35380 

7 

54 

60422 

39578 

62574 

37426 

64654 

35346 

6 

55 

60459 

39541 

62609 

37391 

64688 

35312 

5 

56 

60495 

39505 

62645 

37355 

64722 

35278 

4 

57 

60532 

39468 

62680 

37320 

64756 

35244 

8 

58 

60568 

39432 

62715 

37285 

64790 

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2 

59 

60605 

39395 

62750 

37250 

64824 

35176 

1 

60 

60641 

39359 

62785 

87215 

64858 

85142 

0 

/ 

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68* 

67° 

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158 


LOG.  TANGENTS  AND  COTANGENTS 


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9.64858 

10.35142 

9.66867 

10.33133 

9.68818 

10.31182 

60 

1 

64892 

35108 

66900 

33100 

68850 

31150 

59 

2 

64926 

35074 

60933 

33067 

68882 

31118 

58 

3 

649GO 

35040 

66966 

33034 

68914 

31086 

57 

4 

64994 

35006 

66999 

33001 

68946 

31054 

56 

5 

65028 

34972 

67032 

32968 

68978 

31022 

55 

6 

65062 

34938 

67065 

32935 

69010 

80990 

54 

7 

65096 

34904 

67098 

32902 

69042 

30958 

53 

8 

65130 

34870 

67131 

32869 

69074 

30926 

52 

9 

65164 

34836 

67163 

32837 

69106 

30894 

51 

10 

9.65197 

10.34803 

9.67196 

10.32804 

9.69138 

10.30862 

50 

11 

65231 

34769 

67229 

32771 

69170 

30830    49 

12 

65265 

34735 

67262 

32738 

69202 

80798 

48 

13 

65299 

34701 

67295 

32705 

69234 

30766 

47 

14 

65333 

34667 

67327 

32673 

69266 

30734 

46 

15 

65366 

34634 

67360 

32640 

69298 

30702 

45 

16 

65400 

34600 

67393 

32607 

69329 

30671 

44 

17 

65434 

34566 

67426 

32574 

69361 

30639 

43 

18 

654G7 

34533 

67458 

32542 

69393 

30607 

42 

19 

65501 

34499 

67491 

325C9 

69425 

30575 

41 

20 

9.65535 

10.34465 

9.67524 

10.32476 

9.69457 

10.30543 

40 

21 

65568 

34432 

67556 

32444 

69488 

30512 

39 

22 

65602 

34398 

67589 

30411 

69520 

30480 

38 

23 

65636 

34364 

67622 

32378 

69552 

30448 

37 

24 

656C9 

34331 

67654 

32346 

69584 

30416 

36 

25 

65703 

34297 

67687 

32313 

69615 

30385 

35 

26 

65736 

34264 

67719 

32281 

69647 

30353 

34 

27 

65770 

34230 

67752 

32248 

69679 

80321 

33 

28 

65S03 

34197 

67785 

32215 

69710 

30290 

32 

29 

65837 

34163 

67817 

32183 

69742 

30258 

31 

30 

9.65870 

10.34130 

9.67850 

10.32150 

9.69774 

10.30226 

30 

31 

65904 

34096 

67882 

32118 

t>9805 

30195 

29 

32 

65937 

34063 

67915 

32085 

69837 

30163 

28 

33 

65971 

34029 

67947 

32053 

69868 

30132 

27  ' 

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66004 

33996 

67980 

32020 

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30100 

26 

35 

66038 

33962 

68012 

31988 

69932 

30068 

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66071 

83929 

68044 

31956 

69963 

30037 

24 

37 

66104 

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68077 

31923 

69995 

30005 

23 

38 

66138 

33862 

68109 

31891 

70026 

•  29974 

22 

39 

66171 

33829 

68142 

31858 

70058 

29942 

21 

40 

9.66204 

10.33796 

9.68174 

10.31826 

9.70089 

10.29911 

20 

41 

66238 

33762 

68206 

31794 

70121 

29879 

19 

42 

66271 

33729 

68239 

31761 

70152 

29848 

18 

43 

66304 

33696 

68271 

31729 

70184 

29816 

17 

44 

66337 

33663 

68303 

31697 

70215 

29785 

16 

45 

66371 

33629 

68336 

31664 

70247 

29753 

15 

46 

66404 

33596 

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31632 

70278 

29722 

14 

47 

66437 

33563 

68400 

31600 

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13 

48 

66470 

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68432 

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49 

66503 

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31535 

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11 

50 

9.66537 

10.33463 

9.68497 

10.31503 

9.70404 

10.29596 

10 

51 

66570 

33430 

68529 

31471 

70435 

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9 

52 

66603 

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31439 

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53 

66636 

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31407 

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7 

54 

66669 

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31374 

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6 

55 

66702 

33298 

68658 

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5 

56 

66735 

33265 

68690 

31310 

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4 

57 

66768 

33232 

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3 

58 

66801 

33199 

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31246 

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2 

59 

66834 

33168 

68786 

31214 

70685 

29315 

1 

60 

66867 

33133 

68818 

31182 

70717 

29283 

0 

/ 

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Tan 

Cotan 

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/ 

66° 

64° 

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LOG.   TANGENTS  AND   COTANGENTS 


159 


27° 

28° 

29° 

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Tan 

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9.70717 

10.29283 

9.72567 

10.27433 

9.74375 

10.25625 

60 

1 

70748 

29252 

72598 

27402 

74405 

25595 

59 

2 

70779 

29221 

72628 

27372 

74435 

25565 

58 

3 

70810 

29190 

72659 

27341 

74465 

25535 

57 

4 

70841 

29159 

72689 

27311 

74494 

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5 

70873 

29127 

72720 

27280 

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55 

6 

70904 

29096 

72750 

27250 

74554 

25446 

54 

7 

70935 

29065 

72780 

27220 

74583 

25417 

53 

8 

70906 

29034 

72811 

27189 

7461  3 

25387 

52 

9 

70997 

29003 

72841 

27159 

74643 

25357 

51 

10 

9.71028 

10.28972 

9.72872 

10.27128 

9.74673 

10.25327 

50 

11 

71059 

28941 

72902 

27098 

74702 

25298 

49 

12 

71090 

28910 

72932 

27'068 

74732 

25268    48 

13 

71121 

28879 

72963 

27037 

74762 

25238    47 

!  14 

71153 

28847 

72993 

27007 

74791 

25209 

46 

I  15 

71184 

28816 

73023 

26977 

7'4821 

25179 

45 

i  16 

71215 

28785 

73054 

26946 

74851 

25149    44 

1  17 

71246 

28754 

73084 

26916 

74880 

25120 

43 

18 

71277 

28723 

78114 

26886 

74910 

25090 

42 

19 

71308 

28692 

73144 

26856 

74939 

25061 

41 

20 

9.71339 

10.28661 

9.73175 

10.26825 

9.74969 

10.25031 

40 

21 

71370 

28630 

73205 

26795 

74998 

25002 

39 

22 

71401 

28599 

73235 

26765 

75028 

24972 

38 

-,3 

71431 

28569 

73265 

26735 

75058 

24942 

37 

24 

71462 

28538 

73295 

26705 

75087 

24913 

36 

25 

71493 

28507 

73326 

26674 

75117 

24883 

35 

26 

71524 

28476 

7:3356 

26644 

75146 

24854 

34 

27 

71555 

28445 

73386 

26614 

75176 

24824 

33 

28 

71536 

28.414 

73416 

26584 

75205 

24795 

32 

29 

71617 

28383 

73446 

26554 

75235 

24765 

31 

30 

9.71648 

10.28352 

9.73476 

10.26524 

9.75264 

10.24736 

30 

.'31 

71679 

28321 

73507 

26493 

75294 

24706 

29 

32 

71709 

28291 

73537 

26463 

75323 

24677 

28 

33 

71740 

28260 

73567 

26433 

75353 

24647 

27 

31 

71771 

28229 

73597 

26403 

75382 

24618 

26 

35 

71802 

28198 

73627 

26373 

75411 

21589 

25 

36 

71833 

28167 

73657 

26343 

75441 

24559 

24 

37 

71863 

28137 

73687 

26313 

75470 

24530 

23 

38 

71894 

28103 

73717 

26283 

75500 

24500 

22 

39 

71925 

28075 

73747 

26253 

75529 

24471 

21 

40 

9.71955 

10.28045 

9.73777 

10.26223 

9.75558 

10.24442 

20 

41 

71986 

28014 

73807 

26193 

75588 

24412 

19 

42 

72017 

27983 

73837 

26163 

75617 

24383 

18 

43 

72048 

279.V2 

73867 

26133 

75647 

24353 

17 

44 

72078 

27922 

73897 

26103 

75676 

24324 

16 

45 

72109 

27891 

73927 

26073 

75705 

24295 

15 

46 

72140 

27860 

73957 

26043 

75735 

242C5 

14 

47 

72170 

27830 

73987 

26013 

75764 

24236 

13 

48 

72201 

27799 

74017 

25983 

75793 

24207 

12 

49 

72231 

27769 

74047 

25953 

75822 

24178 

11 

i  50 

9.72262 

10.27738 

9.74077 

10.25923 

9.75852 

10.24148 

10 

51 

72293 

27707 

74107 

25893 

75881 

24119 

9 

52 

72323 

27677 

74137 

25863 

75910 

24090 

8 

53 

72354 

27646 

74166 

25834 

75939 

24061 

7 

54 

72384 

27616 

74196 

25804 

75969 

24031 

6 

55 

72415 

27585 

74226 

25774 

75998 

24002 

5 

56 

72445 

27555 

74256 

25744 

76027 

23973 

4 

57 

72476 

27524 

74286 

25714 

76056 

23944 

3 

58 

72506 

27494 

74316 

25684 

76086 

28914 

2 

59 

72537 

27463 

74345 

25655 

76115 

23885 

1 

60 

72567 

27433 

74375 

25625 

76144 

23856 

0 

/ 

Cotan 

Tan 

Cotan     Tan 

Cotan 

Tan 

, 

62* 

61« 

60° 

100 


LOG.  TANGENTS  AND  COTANGENTS 


30° 

31° 

32° 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.76144 

10.23856 

9.77877 

10.22123 

9.79579 

10.20421 

60 

76173 

23827 

77906 

22094 

79607 

20393 

59 

2 

76202 

23798 

77935 

220G5 

79G35 

20365 

58 

3 

76231 

23769 

77963 

22037 

79663 

20337 

57 

4 

76261 

23739 

77992 

22008 

79691 

20309 

56 

5 

76290 

23710 

78020 

21980 

79719 

20281 

55 

6 

76319 

23681 

78049 

21951 

79747 

20253 

54 

7 

76348 

23652 

78077 

21923 

79776 

20224 

53 

8 

76377 

23623 

78106 

21894 

79804 

20196 

52 

9 

76406 

23594 

78135 

21865 

79882 

20168 

51 

10 

9.76435 

10.23565 

9.78163 

10.21837 

9.79860 

10.20140 

50 

11 

76464 

23536 

78192 

21808 

79888 

20112 

49 

12 

76493 

23507 

78220 

21780 

79916 

20084 

48 

13 

76522 

23478 

78249 

21751 

79944 

20056 

47 

14 

76551 

23449 

78277 

21723 

79972 

20028 

46 

15 

76580 

23420 

78306 

21694 

80000 

20000 

45 

16 

76609 

23391 

78334 

21666 

80028 

19972 

44 

17 

76639 

23361 

78363 

21637 

80056 

19944 

43 

18 

76668 

23332 

78391 

21609 

80084 

19916 

42 

19 

76697 

23303 

78419 

21581 

80112 

19888 

41 

20 

9.76725 

10.23275 

9.78448 

10.21552 

9.80140 

10.19860 

40 

21 

76754 

23246 

78476 

21524 

80168 

19832 

39 

22 

7678S 

23217 

78505 

21495 

80195 

19805 

38 

23 

76812 

23188 

78533 

21467 

80223 

19777 

37 

24 

76841 

23159 

78562 

21438 

80251 

19749 

36 

25 

76870 

23180 

78590 

21410 

80279 

19721 

35 

26 

76899 

23101 

78618 

21382 

80307 

19693 

34 

27 

76928 

23072 

78647 

21353 

80335 

19665 

33 

28 

76957 

23043 

78675 

21325 

8G363 

19637 

32 

29 

76986 

23014 

78704 

21296 

80391 

19609 

31 

30 

9.77015 

10.22985 

9.78732 

10.21268 

9.80419 

10.19581 

30 

31 

77044 

22956 

78760 

21240 

80447 

19553 

29 

32 

77073 

22927 

78789 

21211 

80474 

19526 

28 

33 

77101 

22899 

78817 

21183 

80502 

19498 

27 

34 

77130 

22870 

78845 

21155 

80530 

19470 

26 

35 

77159 

22841 

78874 

21126 

80558 

19442 

25 

36 

77188 

22812 

78902 

21098 

80586 

19414 

24 

37 

77217 

22783 

78930 

21070 

80614 

19386 

23 

38 

77246 

22754 

78959 

21041 

80642 

19358 

22 

39 

77274 

22726 

78987 

21013 

80669 

19331 

21 

40 

9.77303 

10.22697 

9.79015 

10.20985 

9.80697 

10.19303 

20 

41 

77332 

22668 

79043 

20957 

80725 

19275 

19 

42 

77361 

22639 

79072 

20928 

80753 

19247 

18 

43 

77390 

22610 

79100 

20900 

80781 

19219 

17 

44 

77418 

22582 

79128 

20872 

•  80808 

19192 

16 

45 

77447 

22553 

79156 

20844 

80836 

19164 

15 

46 

77476 

22524 

79185 

20815 

80864 

19136 

14 

47 

77505 

22495 

79213 

20787 

80892 

19108 

13 

48 

77533 

22467 

79241 

20759 

80919 

19081 

12 

49 

77562 

22438 

•  79269 

20731 

80947 

19053 

11 

50 

9.77591 

10.22409 

9.79297 

10.20703 

9.80975 

10.19025 

10 

51 

77619 

22381 

79326 

20674 

81003 

18997 

9 

52 

77648 

22352 

79354 

20646 

81030 

18970 

8 

53 

77677 

22323 

79382 

20618 

81058 

18942 

7 

54 

77706 

22294 

79410 

20590 

81086 

18914 

6 

55 

77734 

22266 

79438 

20562 

81113 

18887 

5 

56 

77763 

22237 

79466 

20534 

81141 

18859 

4 

57 

77791 

22209 

79495 

20505 

81169 

18831 

3 

58 

77820 

22180 

79523 

20477 

81196 

18804 

2 

59 

77849 

22151 

79551 

20449 

81224 

18776 

1 

60 

77877 

22123 

79579 

20421 

81252 

18748 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

, 

69° 

68° 

67° 

LOG.   TANGENTS  AND  COTANGENTS 


161 


88° 

84° 

36° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

9.81252 

10.18748 

9.82899 

10.17101 

9.84523 

10.15477 

60 

1 

81-279 

18721 

82926 

17074 

84550 

15450 

59 

2 

81307 

18693 

82953 

17047 

84576 

15424 

58 

3 

81335 

18665 

82980 

17020 

84603 

15397 

57 

4 

81362 

18638 

83008 

16992 

84630 

15370 

56 

5 

81390 

18610 

83035 

16965 

84657 

15343 

55 

g 

81418 

18582 

83062 

16938 

84684 

15316 

54 

7 

81445 

18555 

83089 

16911 

84711 

15289 

53 

§ 

81473 

18527 

83117 

16883 

84738 

15262 

52 

9 

81500 

18500 

83144 

16856 

84764 

15236 

51 

10 

9.81528 

10.18472 

9.83171 

10.16829 

9.84791 

10.15209 

50 

81556 

18444 

83.198 

16802 

84818 

15182 

49 

12 

81583 

18417 

83225 

16775 

84845 

15155 

48 

12 

81611 

18389 

83252 

16748 

84872 

15128 

47 

14 

81638 

18362 

83280 

16720 

84899 

15101 

46 

15 

81666 

18334 

83307 

16693 

84925 

15075 

45 

16 

81693 

18307 

83334 

16666 

84952 

15048 

44 

17 

81721 

18279 

83361 

16639 

84979 

15021 

43 

18 

81748 

18252 

83388 

16612 

85006 

14994 

42 

19 

81776 

18224 

83415 

16585 

85033 

14967 

41 

20 

9.81803 

10.18197 

9.88442 

10.16558 

9.85059 

10.14941 

40 

21 

81831 

18169 

83470 

16530 

85086 

14914 

39 

22 

81858 

18142 

83497 

165015 

85113 

14887 

38 

23 

81886 

18114 

83524 

16476 

85140 

14860 

37 

\  24 

81913 

18087 

83551 

16449 

85166 

14834 

36 

25 

81941 

18059 

83578 

16422 

85193 

14807 

35 

26 

81968 

18032 

83605 

16395 

85220 

14780 

34 

27 

81996 

18004 

83632 

16368 

85247 

14753 

33 

28 

82023 

17977 

83659 

16341 

85273 

14727 

82 

29 

82051 

17949 

83686 

16314 

85300 

14700 

31 

30 

9.82078 

10.17922 

9.83713 

10.16287 

9.85327 

10.14673 

80 

31 

82106 

17894 

83740 

16260 

85354 

14646 

29 

32 

82133 

178«7 

83768 

16232 

85380 

14620 

28 

83 

82161 

17839 

83795 

16205 

85407 

14593 

27 

34 

82188 

17812 

83822 

16178 

85434 

14566 

26 

35 

82215 

17785 

83849 

16151 

85460 

14540 

25 

36 

82243 

17757 

83876 

16124 

85487 

14513 

24 

37 

82270 

17730 

83903 

16097 

85514 

14486 

23 

38 

82298 

17702 

83930 

16070 

85540 

14460 

22 

39 

82325 

17675 

83957 

16043 

85567 

14433 

21 

40 

9.82352 

10.17648 

9.83984 

10.16016 

9.85594 

10.14406 

20 

41 

82880 

17620 

84011 

15989 

85620 

14380 

19 

42 

82407 

17593 

84038 

15962 

85647 

14353 

18 

43 

82435 

17565 

84065 

15935 

85674 

14326 

17 

44 

82462 

17538 

84092 

15908 

85700 

14300 

16 

45 

82489 

17511 

84119 

15881 

85727 

14273 

15 

46 

82517 

17483 

84146 

15854 

85754 

14246 

14 

47 

82544 

17456 

84173 

15827 

85780 

14220 

18 

48 

82571 

17429 

84200 

15800 

85807 

14193 

12 

49 

82599 

17401 

84227 

15773 

85834 

14166 

11 

50 

9.82626 

10.17374 

9.84254 

10.15746 

9.85860 

10.14140 

10 

51 

82653 

17347 

84280 

15720 

85887 

14113 

9 

52 

82681 

17319 

84307 

15693 

85913 

14087 

8 

53 

82708 

17292 

84334 

15666 

85940 

14060 

7 

54 
55 

82735 
82762 

17265 
17238 

84361 
84388 

15639 
15612 

85967 
85993 

14033 
14007 

6 
5 

56 

82790 

17210 

84415 

15585 

86020 

13980 

4 

57 

82817 

17183 

84442 

15558 

86046 

13954 

3 

58 

82844 

17156 

84469 

15531 

86073 

13927 

2 

59 

82871 

17129 

84496 

15504 

86100 

13900 

1 

60 

82899 

17101 

84523 

15477 

86126 

13874 

0 

~f 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

56° 

66° 

64° 

162 


LOG.   TANGENTS   AND   COTANGENTS 


/ 

86° 

87° 

88° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotau 

0 

9.86126 

10.13874 

9.87711 

10.12289 

9.89281 

10.10-719 

60 

1 

86153 

13847 

87738 

12262 

89307 

10693 

59 

2 

86179 

13821 

87764 

12236 

89333 

10667 

58 

3 

86206 

13794 

87790 

12210 

89359 

10641 

57 

4 

86232 

13768 

87817 

12183 

89385 

10615 

56 

5 

86259 

13741 

87843 

12157 

89411 

10589 

55 

6 

86285 

13715 

87869 

12131 

89437 

10563 

54 

7 

86312 

13668 

87895 

12105 

89463 

10537 

53 

8 

86338 

13662 

87922 

12078 

89489 

10511 

52 

9 

863C5 

13635 

87948 

12052 

89515 

10485 

51 

10 

9.86392 

10.13608 

9.87974 

10.12026 

9.89541 

10.10459 

50 

11 

86418 

13582 

88000 

12000 

89567 

10433    49 

12 

86445 

13555 

88027 

11973 

89593 

10407    48 

13 

86471 

13529 

88053 

11947 

89619 

10381    47 

14 

86498 

13502 

88079 

11921 

89645 

10355 

46 

15 

86524 

13476 

88105 

11895 

89671 

10329 

45 

16 

86551 

13449 

88131 

11869 

89697 

10303 

44 

17 

86577 

13423 

88158 

11842 

89723 

10277 

43 

18 

86603 

13397 

88184 

11816 

89749 

10251 

42 

19 

86630 

13370 

88210 

11790 

89775 

10225 

41 

20 

9.86656 

10.13344 

9.88236 

10.11764 

9.89801 

10.10199 

40 

21 

86683 

13317 

88262 

11738 

89827 

10173 

39 

22 

86709 

13291 

88289 

11711 

89853 

10147 

38 

23 

86736 

13264 

88315 

11685 

89879 

10121 

37 

24 

86762 

13238 

88341 

11659 

89905 

10095 

36 

25 

.  86789 

13211 

88367 

11633 

89931 

10069 

35 

26 

86815 

13185 

88893 

11607 

89957 

10043 

34 

27 

86842 

13158 

88420 

11580 

89983 

10017 

33 

28 

86868 

13132 

88446 

11554 

90009 

09991 

32 

29 

86894 

13106 

88472 

11528 

90035 

09965 

31 

30 

9.86921 

10.13079 

9.88498 

10.11502 

9.90061 

10.09939 

30 

31 

86947 

13053 

88524 

11476 

90086 

09914 

29 

32 

86974 

13026 

88550 

11450 

90112 

09888 

28 

33 

87000 

13000 

88577 

11423 

90138 

09862 

27 

34 

87027 

12973 

88603 

11397 

90164 

09836 

26 

35 

87053 

12947 

88629 

11371 

90190 

09810 

25 

36 

87079 

12921 

8H655 

11345 

90216 

09784 

24 

37 

87106 

12894 

88681 

11319 

90242 

09758 

23 

38 

87132 

12868 

88707 

11293 

902(58 

09732 

22 

39 

87158 

12842 

88733 

11267 

90294 

09706 

21 

40 

9.87185 

10.12815 

9.88759 

10.11241 

9.90320 

10.09080 

20 

41 

87211 

12789 

88786 

11214 

90346 

09654 

19 

42 

87238 

12762 

88812 

11188 

90371 

09629 

18 

43 

87264 

12736 

88838 

11162 

90397 

09603 

17 

44 

87290 

12710 

88884 

11136 

90423 

09577 

16 

45 

87317 

12683 

88890 

11110 

90449 

09551 

15 

46 

87343 

12657 

88916 

11084 

90475 

09525 

14 

47 

87369 

12631 

88942 

11058 

90501 

09499 

13 

48 

87396 

12604 

88968 

11032 

90527 

09473 

12 

49 

87422 

12578 

88994 

11006 

90553 

09447 

11 

50 

9.87448 

10.12552 

9.89020 

10.10980 

9.90578 

10.09422 

10 

51 

87475 

13525 

89046 

10954 

90604 

OOH96 

9 

52 

87501 

12499 

89073 

10927 

90630 

09370 

8 

53 

87527 

12473 

89099 

10901 

90656 

09344 

7 

54 

87554 

12446 

89125 

10875 

90682 

09818 

6 

55 

87580 

12420 

89151 

10849 

90708 

09292 

5 

56 

87606 

12394 

89177 

10823 

90734 

09266 

4 

57 

87633 

12367 

89203 

10797 

90759 

09241 

8 

58 

87659 

12341 

89229 

10771 

90785 

09215 

2 

59 

87685 

12315 

89255 

10745 

90811 

09189 

1 

60 

87711 

12289 

89281 

10719 

90837 

09163 

0 

i 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

63° 

52° 

61° 

LOG.   TANGENTS  AND   COTANGENTS 


163 


89° 

40° 

41° 

/ 

Tan 

Cotan 

Tan 

Cotau 

Tan 

Cotan 

o 

9.90837 

10.09163 

9.92381 

10.07619 

9.93916 

10.06084 

60 

1 

90863 

09137 

92407 

07593 

93942 

06058 

59 

2 

90889 

09111 

92433 

07567 

93967 

06033 

58 

3 

90914 

09086 

92458 

07542 

93993 

06007 

57 

4 

90940 

09060 

92484 

07516 

91018 

05982 

56 

5 

90966 

09034 

92510 

07490 

94044 

05956 

55 

6 

90992 

09008 

92535 

07465 

94069 

05931 

54 

7 

91018 

08982 

92561 

07439 

94095 

05905 

53 

8 

91043 

08957 

92587 

07413 

94120 

05880 

52 

9 

910J9 

08931 

92612 

07388 

94146 

05854 

51 

10 

9  91095 

10.08905 

9.92638 

10.07362 

9.91171 

10.  05829 

50 

11 

91121 

08879 

92663 

07337 

94197 

05803 

49 

12 

91147 

08853 

92689 

07311 

94222 

05778 

48 

13 

91172 

08828 

92715 

07285 

91248 

05752 

47 

14 

91198 

08802 

92740 

07260 

94273 

05727 

46 

15 

9J224 

08776 

92766 

07234 

94299 

05701 

45 

16 

91250 

08750 

92792 

07208 

94324 

05676 

44 

17 

91276 

08724 

92817 

07183 

94350 

05650 

43 

18 

91301 

08699 

92843 

07157 

94375 

05625 

42 

19 

91327 

08673 

92868 

07132 

94401 

05599 

41 

20 

9.91858 

10.08647 

9.92894 

10.07106 

9.94426 

10.05574 

40 

21 

91379 

08621 

92920 

07080 

94452 

05548 

39 

22 

91404 

08596 

92945 

07055 

94477 

05523 

38 

23 

91430 

08570  ' 

92971 

07029 

94503 

05497 

37 

24 

91456 

08544 

92996 

07004 

94528 

05472 

36 

25 

9148.! 

08518 

93022 

06978 

94554 

05446, 

35 

26 

91507 

08493 

93048 

06952 

94579 

05421 

34 

27 

91533 

08467 

93073 

06927 

94604 

05396 

33 

28 

91559 

08441 

93091) 

06901 

94630 

05370 

32 

29 

91583 

08415 

93124 

06876 

94655 

05345 

31 

30 

9.91610 

10.08390 

9.93150 

10.06850 

9.94681 

10.05319 

30 

31 

91630 

08364 

93175 

06825 

94706 

05294 

29 

32 

91662 

08338 

93201 

06799 

94732 

05268 

28 

33 

9168S 

08312 

93227 

06773 

94757 

05243 

27 

34 

91713 

08287 

93252 

06748 

94783 

05217 

26 

35 

91739 

08261 

93278 

06722 

94808 

05192 

25 

36 

91765 

08235 

93303 

06697 

94834 

05166 

24 

37 

91791 

08209 

03329 

06671 

94859 

05141 

23 

38 

91816 

08184 

93354 

06646 

94884 

05116 

22 

39 

91842 

08158 

93380 

06620 

'J4910 

05090 

21 

40 

9.91868 

10.08132 

9.93406 

10.06594 

9.94935 

10.05065 

20 

41 

91893 

08107 

93431 

06569 

94961 

05039 

19 

42 

91919 

08061 

93457 

06543 

94986 

05014 

18 

43 

91945 

08055 

93482 

06518 

95012 

04988 

17 

44 

91971 

OS029 

93508 

06492 

95037 

04963 

16 

45 

91996 

08004 

93533 

06467 

'J5002 

04938 

15 

46 

92022 

07978 

93559 

06441 

95088 

04912 

14 

47 

92048 

07952 

93584 

06416 

95113 

04887 

13 

48 

92073 

07927 

93610 

06390 

95139 

04861 

12 

49 

92099 

07901 

93636 

06364 

95164 

04836 

11 

50 

9.92125 

10.07875 

9.93661 

10.06339 

9.95190 

10.04810 

10 

51 

92150 

07850 

93687 

06313 

95215 

04785 

9 

52 

92176 

07824 

93712 

06288 

95240' 

04760 

8 

.',3 

92202 

07798 

93738 

06262 

95266 

04734 

7 

5"i 

92227 

07773 

93763 

06237 

95291 

04709 

6 

55 

92253 

07747 

93789 

06211 

95317 

04683 

5 

r.6 

92279 

07721 

93814 

06186 

95342 

04658 

4 

57 

92304 

07696 

93840 

06160 

95368 

04632 

3 

58 

92330 

07670 

93865 

06135 

95393 

04607 

2 

59 

92356 

07644 

93891 

06109 

95418 

04582 

1 

00 

92381 

07619 

93916 

06084 

95444 

04556 

0 

/ 

Cotan 

Tan 

Cotau 

Tan 

Cotan 

Tan 

, 

50» 

49° 

48° 

164 


LOG.  TANGENTS  AND  COTANGENTS 


/ 

42° 

43° 

44° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.95444 

10.04556 

9.96966 

10.03034 

9.98484 

10.01516 

60 

1 

95469 

04531 

96991 

03009 

98509 

01491 

59 

2 

95495 

04505 

97016 

02984 

98534 

01466 

58 

3 

95520 

04480 

97042 

02958 

98560 

01440 

57 

4 

95545 

04455 

97067 

02933 

98585 

01415 

56 

5 

95571 

04429 

97092 

02908 

98610 

01390 

55 

6 

95596 

04404 

97118 

02882 

98635 

01365 

54 

7 

95622 

04378 

97143 

02857 

98661 

01339 

53 

8 

95647 

04353 

97168 

02832 

98686 

01314 

52 

9 

95672 

04328 

97193 

02807 

98711 

01289 

51 

10 

9.95698 

10.04302 

9.97219 

10.02781 

9.98737 

10.01263 

50 

11 

95723 

04277 

97244 

02756 

98762 

01238 

49 

18 

95748 

04252 

97269 

02731 

98787 

01213 

48 

13 

95774 

04226 

97295 

02705 

98812 

01188 

47 

14 

95799 

04201 

97320 

02680 

98838 

•  01162 

46 

15 

95825 

04175 

97345 

02655 

98863 

01187 

45 

16 

95850 

04150 

97371 

02629 

98888 

01112 

44 

17 

95875 

04125 

97396 

02604 

98913 

01087 

43 

18 

95901 

04099 

97421 

02579 

98939 

01061 

42 

19 

959*6 

04074 

97447 

02553 

98964 

01086 

41 

20 

9.95952 

10.04048 

9.97472 

10.02528 

9.98989 

10.01011 

40 

21 

95977 

04023 

97497 

02503 

99015 

00985 

39 

22 

96002 

03998 

97523 

02477 

99040 

00960 

38 

23 

96028 

03972 

97548 

02452 

'99065 

00935 

37 

24 

96053 

03947 

97573 

02427 

99090 

00910 

86 

25 

96078 

0392-2 

97598 

02402 

99116 

00884 

35 

26 

96104 

03896 

97624 

02376 

99141 

00859 

34 

27 

96129 

03871 

97649 

02351 

99166 

00834 

33 

28 

96155 

03845 

97674 

02326 

99191 

00809 

82 

29 

96180 

03820 

97700 

02300 

99217 

00783 

31 

30 

9.96205 

10.03795 

9.97725 

10.02^75 

9.99242 

10.00758 

30 

31 

96231 

03769 

97750 

02250 

99267 

00733 

29 

32 

96256 

03744 

97776 

02224 

99293 

00707 

28 

33 

96281 

08719 

97801 

02199 

99318 

00682 

27 

34 

96307 

03693 

97826 

02174 

99343 

00657 

26 

35 

96332 

03668 

97851 

02149 

'  99368 

00632 

25 

36 

96357 

03643 

97877 

02123 

99394 

00606 

24 

37 

96383 

03617 

97902 

02098 

99419 

OC581 

23 

38 

96408 

03592 

97927 

02073 

99444 

00556 

22 

39 

96433 

03567 

97953 

02047 

99469 

00531 

21 

40 

9.96459 

10.03541 

9.97978 

10.02022 

9.99495 

10.00505 

20 

41 

96484 

03516 

98003 

01997 

99520 

00480 

19 

42 

96510 

03490 

98029 

01971 

99545 

00455 

18 

43 

96535 

03465 

98054 

01946 

99570 

00430 

17 

44 

96560 

03440 

98079 

01921 

99596 

00404 

16 

45 

96586 

03414 

98104 

01896 

99621 

00379 

15 

46 

96611 

03389 

98130 

01870 

99646 

00354 

14 

47 

96636 

03364 

98155 

01H45 

99672 

00328 

13 

48 

96662 

03338 

98180 

01820 

99697 

00303 

12 

49 

96687 

03313 

98206 

01794 

99722 

00278 

11 

50 

9.96712 

10.03288 

9.98231 

10.01769 

9.99747 

10.00253 

10 

51 

96738 

03262 

98256 

01744 

99773 

00227 

9 

• 

96763 

03237 

98281 

S  01719 

99798 

00202 

8 

58 

96788 

03212 

98307 

01693 

99823 

00177 

B 

64 

96814 

03186 

98332 

01668 

99848 

00152 

6 

65 

96839 

03161 

98357 

01643 

99874 

00126 

5 

56 

96864 

03136 

98383 

01617 

99899 

00101 

4 

57 

96890 

03110 

98408 

01592 

99924 

00076 

3 

58 

96915 

03085 

98433 

01567 

99949 

00051 

2 

59 

96940 

03060 

98458 

01542 

99975 

00025 

1 

60 

96966 

08034 

98484 

01516 

10.00000 

00000 

0 

f 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

, 

47° 

48* 

46° 

LOG.   VERSED  SINES  AND  EXTERNAL  SECANTS    165 
o°  1° 


' 

Log.  Vers. 

D 

.og.  Exsec. 

D 

Log.  Vers. 

I> 

Log,  Exsec. 

D 

f 

0 

1 

2 
3 
4 

—  oo 
2-62642 
3-22848 
3-58066 
3-83054 

60206 
35218 
24987 

—  CO 

2.62642 
3-22848 
3-58066 
3-83054 

60206 
35218 
24987 

6-18271 
-19707 
.21119 
•22509 
•23877 

1435 
1412 
1389 
1368 

6.18278 
-19714 
-21126 
.  -22516 
•23884 

1436 
1412 
1390 
1368 

O 

1 
2 
3 
4 

5 
6 
7 
8 
9 

4-02436 
-18272 
•31662 
-43280 
-53490 

19382 
15836 
13389 
11598 
10230 

4-02436" 
-18272 
•31662 
.43260 
•53491 

19382 
15836 
13389 
11598 
10230 

6-25223 
-26549 
.27856 
•29142 
-30410 

1346 
1326 
1306 
1286 
1268 

6-25231 
•26557 
-27864 
•29151 
•30419 

1347 
1326 
1306 
1287 
1268 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

4-62642 
-70920 
•78478 
.85431 
.91868 

9151 
8278 
7558 
6953 
6437 

4-62642 
•70921 
.78478 
•85431 
.91868 

9151 
8279 
7557 
6952 
6437 

6-31660 
-32892 
•34107 
•35305 
-36487 

1250 
1232 
1214 
1198 
1182 

6-31669 
-32901 
-34116 
-35315 
-36497 

1250 
1232 
1215 
1198 
1182 

1O 

11 
12 
13 
14 

15 
16 
17 
18 
19 

4-97880 
5-03466 
-08732 
-13696 
-18393 

5992 
5605 
5266 
4964 
4696 

4-97861 
5-03466 
.08732 
.13697 
.18393 

5993 
5605 
5266 
4964 
4696 

6-37653 
.38803 
.39938 
.41059 
-42165 

1166 
1150 
1135 
1121 
1106 

6-37663 
-38814 
-39949 
•41070 
-42177 

1166 
1151 
1135 
1121 
1106 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

5-22348 
.27086 
.31126 
•34987 
-38684 

4455 
4238 
4040 
3861 
3697 

5-22849 
.27087 
-31127 
.34988 
.38685 

4456 
4238 
4040 
3861 
3697 

6-43258 
-44337 
.45403 
•46455 
.47496 

1093 
1078 
1066 
1052 
1040 

6-43270 
•44349 
•45415 
•46468 
•47509 

1093 
1079 
1066 
1053 
1040 

30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

5-42230 
-45636 
•48915 
.52073 
.55121 

3545 
3406 
3278 
3158 
3048 

5-42231 
.45638 
.48916 
.52075 
.55123 

3545 
3407 
3278 
3159 
3048 

6-48524 
.49539 
.50544 
.51536 
.52518 

1028 
1016 
1004 
992 
981 

6-48537 
-49553 
•50557 
.51550 
-52532 

1028 
1015 
1004 
993 
982 

25 
26 
27 
28 
29 

.30 

31 
32 
33 
34 

5-58068 
•60914 
-63372 
-66344 
-68937 

2941 
2848 
2757 
2672 
2593 

5-58068 
.60916 
.63674 
.66346 
•68940 

2945 
2848 
2758 
2672 
2593 

6-53488 
.  54448 
.55397 
.56336 
.57265 

970 
960 
949 
939 
929 

6-53503 
.  54463 
•55413 
.56352 
•57281 

970 
960 
950 
939 
929 

30 

31 
32 
33 

34 

35 
36 
37 
38 
39 

5-71455 
.73902 
•76282 
.78598 
.80854 

2518 
2447 
2379 
2316 
2256 

5-71457 
.73904 
.76284 
.78601 
-80857 

2517 
2447 
2380 
2316 
2256 

6-58184 
.59093 
.59993 
.60884 

.61766 

919 
909 
900 
891 
882 

6-58201 
.59110 
.60011 
.60902 
.61784 

919 
909 
900 
891 
882 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

5-83053 
•85198 
.87291 
.89335 
.91332 

2199 
2145 
2093 
2044 
1996 

5-83056 
•85201 
.87295 
.89338 
.91335 

2199 
2145 
2093 
2043 
1997 

6-62639 
.63503 
.64359 
•65206 
.66045 

872 
864 
855 
847 
839 

6-62657 
.63522 
•64378 
.65226 
-66065 

873 
864 
856 
848 
839 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

5-93284 
•95193 
.97061 
5-98890 
6-00680 

1909 
1868 
1829 
1790 

5-93288 
•95197 
•97065 
5-98894 
6-00685 

1952 
1909 
1868 
1829 
1791 

6-66876 
•67700 
.68515. 
.69323 
•70124 

831 
823 
815 
808 
800 

6-66897 
•67720 
.68536 
.69345 
-70145 

831 
823 
816 
808 
800 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

6-02435 
-04155 
.05842 
•07496 
-09120 

1720 
1686 
1654 
1623 

6-02440 
•04160 
•05847 
•07501 
.09125 

1755 
1720 
1687 
1654 
1623 

6-70917 
•71703 
•72482 
•73254 
•74019 

793 
786 
779 
772 
765 

8-70939 
-71725 
-72505 
•73277 
•  74043 

794 
786 
779 
772 
765 

SO 

51 
52 
53 
54 

55 
56 
57 
58 
59 

6-10714 
•12279 
.13816 
•15327 
.16811 

1565 
1537 
1511 
1484 

6-10719 
•12284 
•13822 
•15333 
•16818 

1594 
1565 
1537 
1511 
1485 

6-  74777 
-75529 
•76275 
•77014 
-77747 

758 
752 
745 
739 
733 

6-74802 
•75554 
•76300 
•77040 
•77773 

759 
752 
746 
739 
733 

55 
56 
57 
58 

5fl 

60 

6-18271 

6.18278 

1460 

6  -  78474 

726 

6-78500 

727 

fifi 

D 

Log.  Exsec 

JO 

Log.  Vers. 

JO 

Log.  Exsec. 

D 

166    LOG.   VERSED  SINES  AND  EXTERNAL  SECANTS 

2°  3° 


/ 

Log,  Vers, 

r> 

,og.  Exsec, 

J> 

Log,  Vers, 

J> 

.og,  Exsec. 

1> 

' 

o 

1 

2 
3 
4 

6-78474 
.79195 
•79909 
.80618 
.81322 

721 
714 
709 
703 

6-78500 
•79221 
.79937 
•80646 
.81350 

721 

715 
709 
703 

7-13687 
.14168 
•14646 
.15122 
.15595 

481 
478 
475 
473 

7-13746 
•14228 
.14707 
.15183 
.15657 

481 
479 
476 
474 

0 

1 
2 
3 
4 

5 
6 
7 
8 
9 

6-82019 
•82711 
.83398 
•84079 
•84755 

697 
692 
686 
681 
676 

6-82048 
.82740 
.83427 
•84109 
.84785 

698 
692 
687 
682 
676 

7-16066 
•16534 
.17000 
.17463 
•17923 

475 
468 
466 
463 
460 

7-16129 
.16598 
•17064 
•17528 
-17989 

471 
469 
466 
464 
461 

5 
6 

7 
8 
9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

6-85425 
-86091 
•86751 
-87407 
.88057 
6-88703 
.89344 
.89980 
.90612 
-91239 

670 
665 

655 
650 
646 
641 
636 
631 
627 

6-85457 
.86123 
.86783 
•87439 
.88090 
6-88737 
•89378 
•90015 
•90647 
-91275 

67l 
666 
660 
656 
651 
646 
641 
636 
632 
628 

7-18382 
.18837 
.19291 
.19742 
.20191 
7-20637 
.21081 
.21523 
.21963 
-22400 

458 
455 
453 
451 
448 
446 
444 
442 
440 
437 

7-18448 
.18905 
•19359 
.19811 
-20260 

7-20707 
-21152 
-21595 
-22035 
-22473 

459 
456 
454 
452 
449 
447 
445 
442 
440 
438 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

6-91862 
.92480 
-93093 
•93703 
.94308 

622 
618 
613 
609 
605 

6-91898 
•92516 
•93131 
.93741 
-94346 

623 
618 
614 
610 
605 

7-22836 
.23269 
.23700 
.24129 
-24555 

435 
433 
431 
429 
426 

7-22909 
-23343 
-23775 
-24204 
-24632 

436 
434 
431 
429 
427 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

6-94909 
•9550G 
.96099 
.96688 
-9727? 

601 
597 
592 
589 
584 

6-94948 
•95545 
.96139 
.96728 

-97313 

601 
597 
593 
589 
585 

7-24980 
.25402 
.25823 
•26241 
•26658 

424 
422 
420 
418 
416 

7-25057 
-25485 
-25902 
•26321 
-26738 

425 
423 
421 
419 
417 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 

6-97853 
.98430 
.99004 
6-99573 
7-00139 
7-00701 
.01259 
.01814 
-02366 

.cmi4 

581 

577 
573 
569 
535 
562 
558 
555 
551 
548 

6-97895 
.98472 
.99046 
6-99616 
7-00182 
7  -  00745 
•01304 
-01860 
-02412 
.02960 

581 
577 
574 
570 
566 
563 
559 
555 
552 
548 

7-27072 
•27485 
.27895 
•28304 
-28711 
7-29116 
.29518 
.29919 
•30319 
-30716 

414 
412 
410 
409 
406 
405 
402 
40  i 
399 
397 

7-27153 
.27567 
.27978 
.28387 
-28795 
7-29205 
-29604 
-30006 
•30406 
•3P804 

415 
413 
411 
409 
407 
405 
404 
402 
400 
398 

30 

31 
32 
33 
34 
35 
36 
37 
38 
39 

40 
41 
42 
43 

44 

7-03458 
.03999 
-04537 
-05071 
-05R03 

542 
541 
537 
534 
531 

7-03505 
.04047 
.04585 
.05125 
-05652 

545 
541 
538 
535 
531 

7-31112 
.31505 
.31897 
.32288 
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94 
94 
94 
94 
94 
93 
94 
93 

0 

2 
3 
4 

6 

7 
8 
9 
10 

20 
30 
40 
50 

6 
7i 
8, 

i?; 
IS 

40 
50 

6 
7 
8 
9 
10 
20 
30 
40 
50 

6 
7 
8 
9 
10 
20 
30 
40 
50 

( 

' 
! 

1! 
2( 
3( 
4( 
5( 

103 

10-3 
12-0 
13-7 
15-4 

III 

51-5 
68-6 
85-8 

100 

10-0 
11-6 
13-5 
15-0 
16-6 
33-3 
50-  G 
66-6 
83-3 

97 

9.7 

11.3 
12-9 
14-5 
16-1 
32-3 
48-5 
64-6 
80-8 

94 

9-4 
10-9 
12.5 
14-1 
15-6 
31-3 
47-0 
62-6 
78-3 

91 

*  9. 

:&• 

!S:- 

)30- 
)45. 
)60- 
)75-i 

103 

10-2 
11.9 
13-6 
15-3 
17-0 
34-0 
51-0 
68-0 
85-0 

99 

9-9 

11-5 
13-2 
14-8 
16-5 
33-0 
49-5 
66.0 
82-5 

96 

9-6 

11-2 
12-8 
14-4 
16-0 
32-0 
48-0 
64-0 
80-0 

93 

9.3 
10.  8 
12-4 
13-9 
15-5 
31-0 
46-5 
62-0 
77-5 

90 

L  9-( 
5  10-. 

:  12.  ( 

nil 

330-( 
545-( 
360-( 
j  75-  < 

101 

10-1 

11-8 
13-3 
15-1 
16-8 
33-6 
50-5 
67-3 
84-1 

98 

9-8 
11-4 
13-0 
14-7 
16-3 
32-6 
49-0 
65-3 
82-6 

95 

9-5 

12'e 

14-2 

31^6 
47-5 
63-§ 
79-1 

92 

10.7 
12.2 
13-8 
15-3 
30.6 
46-0 
61-3 
76-6 

o 

o-5 

0-0 
0-0 
0.1 

o-l 

0-1 
0-2 
0.3 
0-4 

8 
_9_ 

13 
14 

16 

16 
17 

13 

11 
20 

21 
22 
23 

24 

8.47795 
.47897 
.47999 
.48101 
.48203 
8-48304 
.48406 
.48507 
.48609 
.48710 

8-49120 
.49225 
.49331 
.49436 
.49541 

8-53721 
•53816 
.53911 
•54007 
•54102 

8-55^3 
•55342 
•55141 
•55539 
•55638 

5 
6 
7 
8 

9 

8-49646 
.49750 
.49855 
.49960 
.50064 

8-54197 
.54291 
.54386 
.54481 
•54575 
8-54670 
.54764 
.54858 
.54952 
•55046 

8-55736 
•55834 
.55933 
-5603: 
-56129 
8-56226 
•56324 
-56422 
-56519 
•56617 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

8-48811 
.48912 
.49013 
.49114 
.49215 

8-50168 
.50273 
.50377 
.50481 
-50585 

8-49315 
.49415 
.49516 
.49616 
-49716 

8-50688 
.50792 
.50896 
.50999 
-51102 

8-55140 
•55234 
.55328 
.55421 
.55515 

8-56714 
-56812 
-56909 
.57006 
•57103 

20 

21 
22 
23 
24 

25 

27 
28 
29 
30 
31 
32 
33 

81 

35 
36 
37 
38 
39 

8-49816 
-49916 
-50015 
.50115 
-50215 
8-50314 
•50413 
.50512 
•50611 
.50710 

8-51205 
.51309 
'•51412 
.51514 
•51617 

8.55608 
.55701 
•55795 
.55888 
•55981 

8-57200 
.57296 
.57393 
.57490 
•57586 

25 
26 
27 
28 
29 
30 
31 
32 
33 
34 

8-51720 
.51822 
.51925 
.52027 
.52129 

8-56074 
-56166 
.56259 
.56352 
•  56444 

8-57682 
•57779 
•57875 
.57971 
-58067 

8-50809 
.50908 
.51006 
•51105 
.51903 

8-52231 
.52333 
.52435 
.52537 

•  526?,^ 

8-565C6 
.56629 
.56721 
.56813 

.56905 

8-58163 
•58259 
.58354 
-58450 
-58546 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

8-51301 
.51399 
.51497 
.51595 
.51693 

8-5274C 
.52841 
•52943 
.53044 
.53145 

8-56997 
.57089 
.57180 
.57272 
.57363 

8-58641 
-58736 
-58832 
.58927 
-59022 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

8.51791 
•51888 
.51986 
.52083 
-52180 

8-53246 
.53347 
•53448 
.53548 
•53649 

8-57455 
.57546 
.57637 
.57728 
.57819 

8-59117 
-59211 
-59306 
-59401 
.59495 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 

8-52277 
•52374 
-52471 
.52568 
•52665 

8.53749 
.53850 
•53950 
.54050 
.54150 

8.57910 
.58001 
.58092 
.58182 
.58273 

8-59590 
•59684 
-59779 
-59873 
-59967 

50 

51 
52 
53 
54 

8-52761 
•52858 
•52954 
•53050 
.53146 

8-54250 
.54350 
.  54449 
.54549 
.54649 

8-58363 
•58453 
.58544 
.58634 
•58724 

8-60061 
-60155 
.60249 
•60342 
.60436 

55 
56 
57 
58 
59 

6018-53242 
'  [Lg.Vers, 

8-54748 

8-58814 

8-60530 

60 

Log.Exs, 

*> 

Lg.  Vers, 

D 

Log.Exs, 

' 

P.P. 

LOG.   VERSED  SINES  AND  EXTERNAL  SECANTS 
16°  17° 


173 


'  Lg.Vers,  J> 


Log.Exs, 


Lg.Vers, 


Log.Exs 


P.P. 


8-58814 
.58904 
.58993 
.59083 
.59173 
-59262 
.59351 
•59441 
•59530 
-59619 


-59708 
•59797 
•59886 
.59974 
60063 
8-60152 
-60240 
•60328 
.60417 
.60505 


8-60593 
•60681 
•60769 
•60857 
-60944 


8-61032 
•61119 
•61207 
.61294 
-6138! 


8-61469 
•61556 
•61643 
.61730 
•61816 


8-61903 
•61990 
.62076 
.62163 

.62249 


8-62336 
•62422 
•62508 
.62594 


8-62766 
.62852 
.62937 
•  6302 
•6310 


8-63194 
•63279 
•63364 
•63449 
-63534 


8-636li 
.6370; 


.63874 
.63959 


8  -  64043 

Lg.Vers 


86 


85 


8-60530 
.6062 
.6071 
.60810 
•60903 


.61089 
•61182 
.61275 
•61368 


8-61460 
.61553 
.61645 
•61738 
.61830 

8.61922 
•62014 
.62106 
.62198 
-62290 


8-62382 
.62474 
.62565 
.62657 
-62748 


8.62840 
.62931 
•63022 
.63113 
.63204 


8-63295 


•63477 
.63567 
.63658 
8-63748 
.63839 
.63929 
.64019 
-64109 


8-64199 
.64289 
•64379 
.64469 
.64559 

8  -  64649 
•64738 
.64828 
.64917 
-65006 


8-65096 
.65185 
•65274 
•65363 
.65452 


8-65541 

•65629 

.65718 

.65807 

.65895 

8-65984 

Log.Exs 


8-64465 
.64549 
•64633 
•64717 
.64801 


-  64884 
•64968 
.65052 
.65135 

-65218 


8-65302 
•65385 

•65468 
.6555] 


8-65717 
•65800 
.65883 
.65965 
-66048 

8-66131 
.66213 
•66295 
.66378 


8.67359 
.67440 
.67521 
•67602 
-67683 

8-67764 
•67845 
.67926 
.68007 
-68087 


8-65984 


l>  Lg.Vers. 


8-6642! 
•6651S 


-66776 
8-66863 
.66951 
.67039 
•67126 
-67213 


8-67301 
•67388 
.67475 
•  6756: 

.67649 


-67736 
•67822 
.67909 
•67996 
-68082 
8-6816Q 
•68255 
.68341 
.68428 
.68514 


8-6860C 
.68686 
.6877 
.68858 
.68944 


8-6902 
.6911 
.69201 
•69286 
•69372 


1-69457 
-69542 
.6962 
.69712 
.69798 


.7005 
.7013 

-70222 


8-70306 
.70391 
.70475 
.7056 
.7064 


88 


88 


87 


87 


85 


85 


8-70728 


10 

11 
12 
IS 
li 
15 
16 
17 
18 
Jl 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 

31 
32 
33 
li 
35 
36 
37 
38 

-Si 
40 

41 
42 
43 
ii 
45 
46 
47 
48 
ii 
0 
51 
52 
53 
5i 
55 
56 
57 
58 
59 


Log.Exs 


60. 


91 

9-1 
10-6 
12- 1 
13-6 
15-1 
30-5 

3046-5  46- 045- 5 

40  62-0  61- i 

50 


93 

C 

710- 

812. 

13 


10 


15-5 


2031-C 


92 

9-2 
10-7 
12-2 
13-8 

15.  r 

30- i 


77-5  76- 

90     89 

0 
10. 

11 

13. 
14- i 
29-6 


10-5 
812-0 
13-5 
10  15-0 


30- C 


75-8 

88 
8-8 

10-2 
11.7 
13-2 
14-6 
29-3 


45-0  44-5144 


4060-0 
5075-0 

87 
8-7 
10.1 
11, 

13- 


59.3J58- 
74-1 


10 


J.W     Xt  •  U     J.:t  •  O 

2029.028-6 
30  43-5  43- C 
40 

50 


58-0 


72-5  71- 


86 

8-6 

10- C 
11-4 
12-1 


14-5  14- S 


57- 


73-3 

85 
85 
9-9 
11-3 
12-7 
14- 1 
28-3 
42-5 
56-6 
70-8 


84     83     82 


10 


40 


11.2 
12-6 
14-0 


2028-0 


30  42-041- 


56-055 


11-0 
12-4 
13-8 
27-6 


5070-1 


81     80 


10-8 
12-1 


10-6 
12-0 

13-5  13-3 
20  27-0  26-6 
3040-540-0 
54-0  53-3 
67-5166-6 


10-8 
12- S 
13- 6 
27- S 
41. C 
54- 6 
68- 1 

79 

7-t 
9. 

10. 
11. 
13- 
26. 
39. 
52. 
65- 


O 

60- S 

70-0 

80-5 

90-1 

100-1 

200-1 

30  0-2 

40  0-3 

Ff  0-4 


P.  P. 


174    LOG.   VERSED  SINES  AND  EXTERNAL  SECANTS 

18°  19° 


' 

Lg.Vers. 

D 

Log.Exs. 

D 

Lg.Vers. 

D 

Log.  Exs 

» 

/ 

P.P. 

o 

1 

2 

3-68969 
.69049 
•69129 

79 
80 

8-71149 
.71232 
.71316 

83 
84 

8.73625 
.73700 
.73775 

75 
75 

8-76058 
.76137 
•76217 

79 
II 

C 

1 
2 

3 

.69208 
-69288 

79 

.71400 
.71484 

84 

•73851 
.73926 

75 

.76297 
•76376 

79° 

3 

4 

84  83  82 

5 

8-69367 

7  A 

8-71567 

83 

8.74001 

Vb 

8-76456 

bu 

7Q 

5 

6 

0.4 

90 

9 

6 
7 

-  69446 
-69526 

79 

7Q 

.71651 
•71734 

83 

.74076 
•74151 

75 

•76536 
•76615 

79 

6 

7 

8 

11-2 

1  9  fi 

192 

10  : 

1  9 

1 

8 
9 

-69605 
•69684 

79 

•71817 
•71901 

83 

•74226 
•74301 

75 

•76694 
•76774 

79 

8 
9 

10 
20 

14-0 
28-0 

13-8 
27-6 

13- 
27. 

! 

10 

11 
12 

8-69763 
•69842 
•69921 

V9 
79 
79 
75 

8-71984 
•72067 
•72150 

83 
83 
83 

00 

8-74376 
.74451 
.74526 

Vb 
74 
75 

8-76853 
•76932 
•77011 

V9 
79 
79 

70 

10 

11 
12 

30 
40 
50 

42-0 
56-0 
70-0 

41-5 
55-3 
69-1 

41.  ( 
54- 
68- 

13 

•7000C 

7Q 

•72233 

•74600 

"7/f 

•7709C 

13 

14 

.70079 

•72316 

•74675 

•77169 

14 

15 
16 
17 

8-70157 
-70236 
-70314 

?! 

8-72399 
•72481 
•72564 

1 

83 

8  •  74749 
.74824 
•74898 

74 
71 

8-77248 
•77327 
.77406 

V9 
79 
78 

15 
16 
17 

6 

81 
8-1 
9-4 

80 

8-0 
9-1 

79 

7-! 

9-: 

) 

18 
19. 
20 

21 
22 
23 
24 

-70393 
•70471 

78 
78 
78 
78 
78 

•72647 

.72729 

82 
82 
82 
82 
82 
82 

•74973 

•75047 

74 
74 
74 
74 
74 
74 

•77485 

-77563 

78 
78 
78 
79 
78 
78 

18 

19 

9 
10 
20 
30 
40 
50 

10-8 
12-1 
13-5 
27-0 
40.5 
54-0 
67-5 

10-6 
12-0 
13-3 
26-6 
40-0 
53.3 
66-6 

1U- 

11.  i 

13-1 
26.; 
39., 

52- 

05.; 

! 

i 

8-70550 
-70628 
-70706 
•70784 
•70862 

8-72812 
•72894 
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•73059 
.73141 

8.75121 
.75195 
•75269 
.75343 
.75417 

8-77642 
-77720 
•77799 
•77877 
-77956 

21 
22 
23 
24 

25 

8  •  70940 

V8 
7ft 

8-73223 

82 

89 

8.75491 

V4 

70 

8-78034 

V8 

25 

•71018 

77 

.73306 

•75565 

•78112 

7H 

78  77  76 

27 

•71096 

7ft 

•73388 

.75639 

70 

•78191 

10 

27 

6 

7-8 

7-7 

7- 

28 
29 

•71174 
•71251 

71 

.73470 
•73551 

81 

.75712 
•75786 

73 

.78269 
-78347 

78 

28 
29 

7 
8 

9-1 
10-4 

9-C 
10-2 

8-i 

10.  i 

30 
31 
32 
33 
31 
85 

8-71329 
•71406 
•71484 
•71561 
•71639 

77 
77 
77 
77 
77 

8-73633 
•73715 
•73797 
.73878 
.73960 

82 
82 
81 
81 
81 
81 

8-75860 
•75933 
.76006 
.76080 
.76153 

74 
73 
73 
73 
73 
73 

8-78425 
•78503 
.78581 
.78659 
•78736 

V8 
78 
78 

78 

77 

30 

31 
32 
33 
34 

9 
10 
20 
30 
40 
50 

11.  V 
13-0 
26-0 
39-0 
52-0 
65-0 

11-  b 
12-8 
25-6 
38-5 
51-3 
64-1 

11.  ' 

12-1 
25-C 
38-  ( 
50-  i 
83-  J 

8-71716 

8  .  74041 

8-76226 

8-78814 

35 

36 

-71793 

77 

•74123 

8T 

.76300 

70 

•78892 

36 

c-ooc- 
oocooo 

•71870 
•71947 

•72024 

77 
77 

.74204 
•74286 

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81 
81 

.76373 
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•76519 

73 
73 

•78969 
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77 

37 
38 
39 

6 
7 

75 
7-5 
R.7 

74 

7-4 
8-fi 

73 

7-3 
8-5 

40 

8-72101 

U 

8  -  74448 

81 

8T 

8-76592 

73 

8-79202 

yy 

40 

8 

10.0 

9-8 

9. 

41 
4?, 

•72178 
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77 

.74529 
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1 

81 

.76664 
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73 

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41 

9 
10 

11-2 
12-5 

11-1 
12-3 

10-8 
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43 
44 

•72331 
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77 

.74691 
•74772 

81 
80 

.76810 
•76883 

73 

•79434 
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77 
77 

43 
44 

20 
30 

25-0 
37-5 

24-6 
37-0 

24- 
36- 

45 
46 

8-72485 
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Vb 
76 

8-74853 
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81 
81 

pri 

8-76955 
•77028 

V2 
72 

8-79588 
•79665 

yy 

77 

45 
46 

40 
50 

bU.O 
62-5 

49-3 
61-6 

48-  1 
60.  j 

47 

-72637 

.75014 

on 

.77100 

.79742 

77 

47 

48 
49 

-72714 
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76 

.75095 
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80 

.77173 
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72 
72 

.79819 
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77 
77 

48 
49 

72  71  <5 

50 

8-72866 

Vb 
7fi 

8-75256 

8U 
an 

8-77317 

V2 

8-79973 

Vb 

50 

' 

'  8-4 

R.fl 

O.fl 

bi 

•72942 

7ft 

•75336 

8n 

.77390 

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51 

i 

9-6 

9-4 

0  f 

52 

•73018 

70 

•75417 

on 

.77462 

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52 

< 

10-  J 

10.fi 

n.i 

53 
54 

•73094 
•73170 

76 

•75497 
•75577 

80 

.77534 
•77606 

72 

.80203 
.80280 

75 

53 
54 

1( 

2f 

12.  C 

24.  r 

11-8 
23-fi 

0-1 

o-I 

55 
56 
57 

8.73246 
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Vb 
76 
75 

8-75658 
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8(5 
80 
80 

8-77678 
.77750 
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V2 
72 

7? 

8-80356 
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Vb 
76 
76 

55 
56 
57 

3C 
4C 
5C 

36.  C 
48.  C 
60-  C 

35-5 
47-  c 
59-1 

0-2 
0.3 
0-4 

58 
59 

•73473 
•73549 

76 

.75898 
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80 

•77893 
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J2 

•80586 
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76 

58 
59 

60 

8.73635 

7b 

8-76058 

8U 

8-78037 

VI 

8-80738 

Vb 

BO 

Lg.  Vers. 

JJ 

Log.Exs. 

D 

Lg.Vers. 

D 

Log.Exs. 

D 

P.  P. 

LOG.   VERSED  SINES  AND  EXTERNAL  SECANTS    175 
20°  21° 


' 

Lg.Vers 

x> 

Log.Exs 

J> 

Lg.Vers 

1> 

Log,  Exs, 

1> 

I 

P.  P. 

0 

1 

2 

8-78037 
•7810J 
•78180 

7\ 

8-80738 
-8081' 
•80891 

76 

76 

8-82229 
.82297 
.82366 

SI 

8-85214 
•85287 
-85360 

73 
73 

O 

2 

3 
4 

•  78251 
•  78323 

71 

•80967 
•81043 

76 

.82434 
•82502 

68 
68 

-85433 
-85506 

72 
73 

3 

ft  ?6  ?5 

74 

5 

7 
8 
9 

8-78394 
•78466 
•78537 
•78608 
•78679 

7* 

77i 
71 

8-81119 
.81195 
.81271 
•81346 
•81422 

V6 
76 
76 
75 
76 

8-82569 
•82637 
.82705 
.82773 
•82841 

6V 
68 
68 
67 
68 

8-85579 
-8565: 
-85724 
•85787 
-85869 

1 

6 

8 
9 

6 
7 
8 

9 
10 
20 

•/•6 
8-8 
1C-  1 
11-4 
12-6 
25-3 

7-5 
8-7 
10-0 

11..  S 

12-5 
25-C 

8. 
9. 
11- 

12- 
21- 

H 

n 

12 

8-78750 
•78821 
•78892 

VI 
71 
71 

8.81498 
•8157J 
.81649 

76 

7E 

8  .  82908 
•82976 
•83043 

b'i 

62 
62 

8.85942 
-86014 
-86087 

7* 

?i 

10 

11 
12 

30 
40 
50 

38-C 
50-6 
ftfl.5 

37-5 
50-0 
62-5 

37- 
49- 
fil. 

13 
ii 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 

•78963 
•79034 

70 

71 
70 

70 
70 

I 

70 

.81725 
•81800 

75 
75 
75 
75 
75 
75 
75 
75 
75 
74 

-83111 
-83178 

67 
62 
67 
67 
67 
67 
67 
67 
67 
67 

-8615S 
.86231 

72 
72 

75 
72 
72 

72 

13 
J4 

15 
16 
17 
18 
.Ii 
20 
21 
22 
23 
24 

9 

8 

9 
10 
20 
30 
40 
50 

73 

7-S 
8-5 
9-7 
10-  S 
12-1 
24  S 
36-5 
48-6 
60.8 

73  71 

7-2  7-] 
8-4  8. 
9-6  9- 
10-810- 
12-011- 
24-023- 
36-  035-  i 
48-047-; 
60.059.1 

8-79105 
•79175 
•79246 
•79317 
•79387 

8-81876 
•81951 
.82026 
•82102 
•82177 

8  -  83246 
-83313 
-83380 
•  8344"? 
•83515 

8  .  86304 
•86376 
-8644! 
-8652i 
.86592 

8-79458 
•79528 
•79598 
•79669 
•79739 

8-82252 
•82327 
•82402 
•82477 
.32552 

8-83582 
•83649 
•83716 
•83788 
-83850 

8-86663 
•  8673 
-8680: 
-8688i 
-P6952 

25 

8-79809 

70 

8-82627 

Vb 

8-83916 

bb 

8  -  8702| 

25 

26 
27 
28 

•79879 
•79949 
•80019 

.70 
70 
70 

•82702 
.82776 
.82851 

« 

.83983 
.84050 
•84117 

ei 

.8709 
.87167 
.87239 

72 
71 
7T 

26 
27 
28 

6 

70 

7-0 
8-1 

69 

6-9 
8-0 

68 

6-8 
7-8 

1 

29 

.80089 

70 

•82926 

74 

.84183 

05 

•87316 

29 

8 

9-3  92 

9  C 

» 

30 

31 
32 
33 
34 

8-80159 
•80229 
•80299 
•80369 
•80438 

VO 

11 
11 

8.83005 
.83075 
.83149 
-.83224 
-83298 

V4 
74 
74 
74 
74 

8.84250 
.84316 
-84383 
-  8444J 
84515 

bb 

ee 

66 
66 
66 

8-87382 
.87453 
.87525 
.8759! 
.87668 

71 

7; 
7; 
7; 

71 

30 

31 
32 
23 
34 

9 

10 
20 
30 
40 
Bfl 

10.510.3110-2 
11.611-5  11  3 
23-323.022  6 
35-C34  5  34  0 
46-646-045  3 

KQ  SlKI  K  KK  K 

35 

8-80508 

69 

8.83373 

74 

7  A 

8-84582 

66 

8-87739 

71 

35 

Kfi 

.80577 

.83447 

7T 

•  84648 

.87811 

71 

36 

37 
3!) 
^(l 

-80647 
•80716 
•80786 

69 

•83521 
•83595 
•83670 

74 
74 

•8478( 

66 
66 

.87881 
.87953 
•P8024 

7! 
71 

37 
38 

6 

67 

6-7 

66 

6-6 

65 

65 

ILL 
40 
43 
42 
43 

8-80855 
•80924 
•80993 
•81063 
•81132 

69 
69 
69 
69 
69 
69 
69 

8.83744 
.83818 
.83892 
•83966 
•8403S 

8-84115 
.84187 

74 
74 
74 

74 

73 

8-84912 
•  8497i 
-8504 
•  851li 
-8517i 

66 
66 
66 
66 
66 
65 
66 

8-88095 
•88166 
•88237 
-88301 
•PP378 

71 
71 
71 
71 
70 
71 
71 
76 

40~ 

41 
42 
43 
44 

8 
9 
10 
20 
30 
40 
50 

8-9 
10-0 
11.1 
22-3 
33-5 
44.6 
55-8 

8-8 
9.9 
11-0 
22-0 
33-0 
44-0 
55-0 

8  6 
97 
10.  f 
21  f 
32-  £ 
43-  1 
64.1 

45 
46 

8.81201 
•81270 

8  -  85242 
-85308 

8  -  88449 
-88526 

45 
46 

47 

•81339 

69 

fio 

•84261 

/.f 

-8537S 

6K 

•88591 

70 

4V 

48 

49 

•81407 
-81476 

69 

•84331 
.84408 

75 

-85439 
-85505 

66 

-88661 
-88*32 

71 

48 
49 

60.  H 

5O 

8-81545 

65 

8.84481 

Q 

J-  855*6 

6b 

1  .88°03 

7fi 

50 

7 

0-fi 

6] 

•81614 

69 

•84555 

7 

-85626 

.888*3 

7fi 

bl 

8 

0-0 

52 

•81682 

68 

.84628" 

73 

•85*0i 

88944 

'y 

b'2 

< 

0-1 

es 

54 

•81751 
•81819 

68 
68 

.  84*02 
.84775 

73 

•85*66 
.85832 

65 

•89014 
-89085 

70 

53 
54 

1C 
2C 

0-1 
0-1 

55 
56 
57 

8.81888 
•81956 
•82025 

65 
68 

8  .  84845 
.84922 
.84995 

73 
73 

7735 

8-85897 
.85962 
.86027 

65 
65 
65 

ce 

-89155 
•89225 
•89?95 

?0° 

55 
56 
57 

3G 
40 
50 

u.g 

0-3 
0.4 

fifi 

•82093 

.85065 

no 

•86092 

65 

.89366 

70 

b8 

59 

.82161 

.85141 

.86158 

.89436 

59 

60 

8-82229 

68 

8-85215 

73 

8-86223 

65 

•89506 

70 

6O 

i 

Lg.Vers. 

D 

Log.Exs. 

/> 

Lg.Vers, 

D 

Log.Exs. 

D 

' 

P.  P. 

176 


LOG.   VERSED  SINES  AND  EXTERNAL  SECANTS 

22°  23° 


Lg.Vers.  -»  Log.Exs,  2>  Lg.Vers,  D  Log.Exs.  J> 


8-86223 
•8628' 


.8641' 
.8648! 


8-86547 
.86612 
.86676 
.86741 
.86805 


8-86870 
.86934 
•86999 
-87063 
.87127 

B~87192 
•87256 
•87320 
•87384 
•87448 

8-87512 
•87576 
•87640 
•87704 
•87768 


8-87832 
•87895 
•87959 
•88023 


8- 88150 
•88213 
•88277 
•88340 

•  88404 
8-88467 

•88530 
•88593 
•88656 
.88720 
8-88783 
•88846 
•8890 

•  8897 
•8903 


8-C9097 
.89160 
•  8922 
•8928 


8-89411 
•89473 
•8953 
•  8959 
•89660 


8-8972 
•  8978 


•89910 
•89972 


8-  00034 


Lg,  Vers 


B- 89856 
.8992 


•90065 
.90135 


3.90205 
•90274 
•90344 
•90413 
.90483 

3-90552 
.90622 
•90691 
•90760 
.90830 


3-90899 
•90968 
.91037 
•91106 
.91175 


3-91244 
•91313 
•91382 
•91451 
•91520 


3.91588 
•91657 
•91726 
•91794 
•91863 


3-91932 
•92000 
.92068 
.92137 
.92205 


B. 92274 

•92342 

.92410 

•92478 

•92546 

92615 

92683 

92751 

92819 

92887 

8.92955 

.93022 

.93090 

•93158 

.93226 


8-93293 


.93429 
.93496 
•93564 


8-93831 


Log.Exs 


68 


67 


8.90034 
•90096 
.90158 
.90220 
.90282 

8.90344 
.90406 
•90467 
•90529 
.90591 

8-90652 
•90714 
•90776 
•90837 


•91021 
•91083 
•91144 

.91205 


3-91876 
-91937 
•91997 
•92058 
•92119 

8-92179 
-92240 
-92300 
•92361 
.92421 


8-92487 
.92542 
•92602 

•  9266 
.9272 

8-9278 
-9284 
-9290 
-9296 
Z.93Q2: 

8-93082 
-93142 

•  9320 

•  9326 

•  9332 
8-9338 

-9344 

•  9350 
.9356 
.9361 

8-93679 
Lg.Vers. 


6! 


8-93631 
.93699 
.93766 
.93833 
•93901 


8-93968 
•94035 
•94102 
•94170 
•94237 


-94304 
•94371 
•94438 
•94505 
•94572 


8 • 94638 
•94705 
•94772 
•94839 
•94905 


8-94972 
.95039 
•95105 
•95172 
^95238 

8  -  95305 
•95371 
-95437 
•95504 
•95570 


•95636 
-95703 
-95769 
-95835 
•_95_901 
•95967 
•96033 


•96165 
•96231 


Log.bxs. 


P.P. 


70 


50 


9- 
10- 
11. 
23- 
35- 


40  46  • 


58. 


67 

6-7 

7-8 

8-9 

10- Q 

11-1 

22-3 

33-5 

44." 


46-0  45-; 
57-5  56- i 


68 

6-8 
7-9 
9-0 

10-2 
il-3 
22-6 
34-0 


50  55-8 


66 

6-6 
7-7 


11-0 


65 

6-5 


10-! 


10 


10. i 


2021-3 
30  32-0 
4042-6 
50J53-3 


61 

6-1 
7-1 


1010.' 
2020.1 
30  30- 
40  40  •  i 
50  50- i 


22-021.1 
33-0  32-! 
44-043-: 
55-0  54.: 


63     62 

6-2 
7-3    7-2 


10-5  10-3 
21.020-6 
31-531-0 
42-041-3 
52-5  51-6 


60 

6-0 

7-0 

8-C 

9-0 

10-0 

20- C 


30-029.! 
40-0  39-; 
50-049.: 


59 

5-9 


19 


6 

60-5 

70-0 

80- (5 

90.1 

1004 

200-1 

300-g 

400-3 

500.4 


P.  P. 


LOG.   VERSED  SINES  AND  EXTERNAL  SECANTS    177 

24°  25° 


'    Lg.Vers.   I> 


5.93679 
•93738 
.93797 
•93857 
•93916 


5-93975 
•94034 
.94094 
•94153 
•94212 


5-94271 
•94330 
•94389 
. 94448 
.94506 

5-94505 
•94624 
.94683 
•94742 
.94800 


5-94859 
•94917 
•94976 
•95034 
.95093 


•9515! 
.95210 
•95268 
•95326 
-95384 


8-95443 
•95501 
•95559 
•95617 
-95675 


8-95733 
-95791 
•95849 
•95907 
.95965 


8-96023 
•96080 
•96138 
•96196 
•96253 


8-96311 
•96368 
•96426 
•96483 
^96541 

8-96598 
•96656 
•96713 
•96770 
•96827 


8-96885 
•96942 
•96999 
.97056 
•97113 


8-97170 


Lg.Vers. 


Log.Exs. 


8-97606 
•97671 
•97736 
•97801 
•97865 


8-97930 
.97995 
.98060 
•98125 
.98190 


8-98254 
•98319 
•98383 
•98448 
•98513 


8-98577" 
•98642 
•98706 
•98770 
-98835 


•98963 
•99028 


•  9915J 


9-00051 
-00114 


9-00178 
•00242 
.00305 
-00369 
•00432 


•00495 

.0055 

-0062 

-00686 

JD0749 

.GOBI 

-0087 

-0093 

-01002 

.01065 

9-01128 

-01191 

-01254 

-01317 

-01380 

9.01443 

Log.Exs. 


64 


Lg.Vers 


8-98020 
•98077 
•98133 
•98190 
-9824 


•  9835 

•  9841 
-98470 
-98527 


8-98583 
-98639 


-98751 


8-98802 
-98918 
-98974 


.99085 
8-99141 
-99197 
•  9925 
•99308 
.99363 


8-99419 
•99474 
•99529 
-99585 
-99640 


8-99695 
-99751 


9991 


9-0002 
-0008 
-0013 
.0019 

9-00246 

-00301 

-00356 

-00411 

.00466 

9 . 00520 

Lg.  Vers. 


Log.Exs. 


•01443 
•01505 
.01568 
-01631 
-01694 


-01756 
-01819 
-01882 
-01944 
-02007 


•02070 
•02132 
•02195 
•02257 
-02319 


9-02382 
•02444 
•02506 
-0256S 
-02631 
-C269S 
•02755 
-02817 
-0288C 
•02942 


9-03004 
-03066 
-03128 
-03190 
-03252 


9-03313 
-03375 
-03437 
•03499 
^03561 

9.C362~2 
•03684 
•03746 
•03807 


9-03930 
•0399_ 
•04051 
•04115 
-04176 


9-04238 
-0429S 
-0436C 
•04421 
-04483 

9-04544 
•04605 
•04666 
•0472? 
-0478F 


•04911 
-0497 
-0503 
•  0509 


Log.Fxs 


63 


61 


61 
61 
61 
61 
61 
61 
61 
60 
61 
~D 


P.P. 


63 


7-1 


62     61 


7. 


19. 
29. 
39. 


5049- 


!: 


28-5 
38.0 
47-5 


30  28. 

40 

5046- 


P.  P. 


178    LOG. 


VERSED  SINES  AND   EXTERNAL  SECANTS 

26°  27° 


' 

Lg.Vers 

D 

Log.  Exs 

1> 

Lg.Vers. 

V 

Log.  Exs 

& 

' 

P.P. 

0 

1 

9-00520 
.00575 

01 

9-05154 
•05215 

61 

9-03740 
•03792 

52 

5n 

9-08752 
-08811 

59 

O 

1 

J 

•00630 
•  0068^ 
•00739 

ft 

•05276 
•05337 
.05398 

65 
61 

•03845 
•03898 
•03950 

53 
52 

•08870 
•08929 
•08988 

59 
59 

2 
3 
4 

61  60  59 

5 
6 

9-00794 
-00848 

bb 
GI 

9-05458 
-05519 

6U 
61 

9-04002 
•04055 

si 

9-09047 
-09106 

59 
59 

5 
6 

6  6- 
7  7- 

80 

1  b-0  5 
7-0  6 

8 

J 

9 

-0090 
-0095 

.oior 

9-0106 

ft 

54 

57 

-05580 
•  0564*5 
•05701 

60 
60 

61 
fin 

•04107 
•04160 
•04212 

52 
52 
52 

-09164 
•0922J 

•09282 

58 
59 
59 
58 

7 
8 
9 

9  9- 
1010- 
20  20- 
3030- 

9-0  8 
10-0  9 
20-019 
5  30-029 

10 

9-05762 

9.04264 

9-09341 

1O 

1] 

•  0112 

•05822 

fin 

•04317 

•09400 

59 

40  40- 

6  40-039 

ll 

•01174 
•  0122 

55 

•05883 
•05943 

•04369 
•04421 

52 

•09458 
•09517 

58 
59 

12 

5050. 

850-049 

I/ 

•01283 

•06004 

60 

•04473 

•09576 

58 

14 

15 
16 

9-0133 
•0139' 

54 
54 

9-06064 
•06124 

bU 
6Q 

9-04525 
•04577 

52 

9-09634 
•09693 

58 
58 

15 

16 

58  57 
6!  5-8  5.| 

17 
18 

ii 

•01445 
•01499 
•01554 

it 

•06185 
•06245 
•06305 

60 

•04630 
•04682 
•04734 

52 
52 

.09752 
•09810 
.09869 

59 
58 
58 

17 
18 
19 

7  6-Z 
8  7-7 
9  8-7 

B-fl 
8  3 

24 

9-01608 
•01662 
•01715 
•01769 
.01823 

ft 

54 
54 

9.06366 
•06426 
•06486 
•06546 
•06606 

60 

S8 

60 
60 

9-04786 
•04837 
•04889 
•04941 
•04993 

li 

52 
52 
52 

9-09927 
•09986 
.  10044 
•10102 
•10161 

58 
58 
58 
58 

21 
22 
23 
24 

IU 
20 
30 
40 
50 

o-o 
19-3 
29-0 
38-6 
48-3 

19.  Q 
28-5 
38.0 
47-3 

9.01877 

54 

50 

9-06667 

ttU 
fin 

9-05045 

51 

9-10219 

58 

2C 

.0193: 

•06727 

•05097 

rf 

•10278 

26 

55  54 

27 

•01985 

j?f 

•06787 

fin 

•05148 

•10336 

08 

27 

6 

5-5 

5.4 

I 

•02038 
•02092 

54 

-06847 
-06907 

60 

•05200 
•05252 

52 

•10394 
•10452 

58 

28 
29 

7 
8 

6-4 
7-3 

8.3 

30 

31 

33 
34 

II 

9-02146 
•02199 
•02253 
•02307 
•02360 

II 
1 

9-06967 
.07027 
.07087 
.07146 
-07206 

bO 
60 

60 
60 
59 

9-05303 
-05355 
-05407 
•05458 
•05510 

51 
52 
51 
51 
51 
51 
51 

9-10511 
•10569 
•10627 
•10685 
•10743 

58 
58 
58 
58 
58 
58 
58 

30 

31 
32 
33 
34 

9 
10 
20 
30 
40 
50 

8-k, 
9-3 
18-  1 
27-5 
36-  1 
15.8 

18.  'O 

27.0 
38.0 
43.9 

9-02414 
•02467 

9-07266 
•07326 

-05561 
•05613 

9-10801 
•10859 

35 

3R 

37 
38 
39 

•02521 
•02574 
•02627 

53 
53 

•07386 
.07445 
•07505 

59 
60 

•05664 
•05715 
-05767 

a 

.10917 
•10975 
•11033 

58 
58 
58 

37 
38 
39 

6 
7 

53 

5-3 

6-? 

5. 

6- 

40 

9-02681 

CO 

9-07565 

K 

•05818 

bl 

9-11091 

b8 

4O 

8 

7-0 

6-? 

41 

•02734 

ii 

•07324 

j?s 

•05869 

51 

•11149 

41 

9 

7-9 

7- 

42 
43 
44 

•02787 
•02840 
•02894 

fi 

•07684 
•07743 
•07803 

60 

•05921 
•05972 
•06023 

a 

•11207 
•11265 
•11323 

57 
58 
58 

42 
43 
44 

10 
20 
30 

8-8 

L7-6 
26-5 

8. 
17. 
26- 

45 
46 

•02947 
•03000 

53 
53 

•07863 
•07922 

by 

59 

•06074 
•06125 

51 

51 

9-11380 
•11438 

b'/ 
58 

45 
4fi 

40, 
50' 

J5-3 

1:4.1 

34- 
43- 

47 
48 
49 

•03053 
•03106 
•03159 

53 
53 

•07981 
•08041 
•08100 

59 
59 

•06176 
•06227 
•06279 

51 
51 
51 

•11496 
•11554 
•11611 

II 
57 

47 
48 
49 

51  0 

50 

51 
52 

•03212 
•03265 
•03318 

bd 
53 
53 
rq 

•08160 
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.18659 

A8 

.79371 

.21609 

47 

48 
49 

•78251 
•78270 

19 

.18708 
.18757 

49 

.79390 
.79409 

18 
19 

.21659 
•21709 

49 

48 
49 

IS 

ft  l  5 

50 

51 
52 

9-78289 
•78309 
•78328 

19 
19 
19 

1  Q 

10-18805 
.18854 
.18903 

48 
48 
49 

AQ 

9.79427 
.79446 
.79465 

18 
19 
19 

10-21759 
.21808 
.21858 

bU 
49 
50 

50 

51 
52 

'/  2.1 
8  2-4 
9  2-8 

53 

•78347 

in 

.18951 

.79484 

.21908 

oy 

58 

10  3-1 

54 

.78366 

.19000 

•79503 

.21958 

54 

20  64 

55 
56 
57 

9-78385 
•  78404 
•  78423 

19 
19 
19 

1  Q 

10.19049 
.19098 
.19146 

48 
49 
48 

AQ 

9.79521 
.79540 
.79559 

18 
18 

10-22008 
•22058 
.22108 

bU 
50 
50 

55 
56 
57 

30  9-2 
40  12-3 
5015.4 

58 

.  78442 

\l 

.19195 

.79578 

.22158 

58 

59 

•78462 

.19244 

.79596 

•22208 

50 

59 

60 

9-78481 

19 
D 

10-19293 

48 

9-79R15 

19 

10.^22258 

bO 

60 

Lg,  Vers, 

Log.Exs. 

•° 

Lg.Vers. 

Log.Exs. 

** 

* 

P.  P. 

LOG.   VERSED  SINES  AND  EXTERNAL  SECANTS 

68°  69° 


199 


'  Lg.Vers.  l> 


18 


18 


18 


Log.Exs,    -»  Lg.Vers. 


10-22258 
.22308 
.22358 
.22408 
•22458 


10-22508 
.2255 
.2260 
.2265 
.2270 


10-22759 
.22809 
.22859 
•22909 
-22960 


LO-2301 
.230 
.2311 
.2316 
.2321 


10.2326: 
.2331! 
.2336! 
.2341! 
-2346; 


0-2351' 
.2356; 
.2361! 
.23661 
•  237li 


10-2376^ 
.2381: 
.2386! 
.2391! 
-2396! 


10  -  24020 
.24071 
.2412 
.2417 
.2422 


10-2427 
.24325 
.24376 
.24427 
.24478 


10-24529 
.24580 
.24631 
.24682 
.24733 


10  -  24784 
.24835 
.24886 


.24988 
10- ^5039 
.25090 
.25142 
.25193 
.25244 


10.25295 

~ 


35295 
.Exs, 


i.  80728 
•80747 
.80765 
•80783 
-80802 


-80820 
•80839 
•80857 
-80875 


9-80912 
.80930 


9-81003 
.81022 
.81040 
.81058 
•81077 

9-81095 
.81113 
.81131 
.81150 
-81168 

9-81186 
. 81204 
.81223 
.81241 
81259 
t.8127 
.81295 
.81314 


9.81361 
•81381 
.81405 
.81423 


9  -8145 
8147 
8149 
8151 
81532 
81550 
.81568 
.81586 


.81640 
.81658 
.81676 
.81695 
.81713 


9-81731 
.81749 
.81767 
.81785 
-81803 


r.Vers. 


Log,  Exs.  ,1> 


10-25295 
-25347 
-25398 
.25449 
.25501 

10".25552 
-25604 
-25655 
•25707 
•25758 


10-2581 
•  2586 
•25913 
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•2S016 


10-2606 
-2611 

•  2617 

•  2622 
-2627 


10-22627 
.27680 
.27732 
-27785 
.27837 


10-2789 
.2794 
.27995 
.28047 
.28100 


10-28152 
•28205 
.28258 
-28310 
.28363 


0-28416 


Log, Exs,    I> 


P.P. 


53 

5-3 


17.1 


10 
20 
3026-5 
40  35-3 
60)44. 1 

52 


3C26. 
4034. 
5043- 


4C 


51 

5-1 
5-9 


17-0 


8025-525 


34  033 


52 

5-2 

6-1 

7-0 

7-9 

8-7 

17-5 

26-2 


7-7 
86 

17-1 
25-7 
34-3 
42-9 

50 
5-B 
5-9 
6-7 
7-6 


5042-542-1 


50 

5-0 


16- 


3025-0 
4033-3 
5041-6 

19     1! 

1-9 


2-5 
2-5 
3-1 
6-3 


4012-i 
5015-i 


12-5 
15-4 


18 
18 

2.1 
2-4 
27 
3-0 
6-0 
0 


40  12-0 
50  IB  0 
P,  P, 


200    LOG.   VERSED   SINES  AND  EXTERNAL  SECANTS 


70° 


71' 


' 

Lg.Vers. 

D 

Log.  Exs. 

n 

Lg.Vers. 

B 

Log,  Exs, 

D 

r 

P 

,  P, 

0 

2 
3 
4 

9-81821 
.81839 
•81857 
.81875 
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18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 
18 

if 

18 
18 
18 
18 
18 
18 

11 

18 
18 
18 

if 

18 
18 
18 
17 
18 

18 

if 

10-28416 
.28469 
•28521 
.28574 
.28627 

53 
53 
52 
53 
53 
53 
53 
53 
53 
53 
53 
53 
53 
53 

II 

53 
53 

53 
53 
53 
53 
53 
53 
53 
53 

53 

11 
it 

II 

54 
54 
54 
54 
54 
54 
54 
54 
54 

i! 

54 
54 
55 

1 

54 
54 

9-82894 
.82911 
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-82947 
-82964 

il 
if 

18 

i! 

17 
18 
17 
17 
17 
18 

17 
18 
17 
17 
17 
17 
18 
17 
17 
17 
17 
17 

18 
17 
17 
17 
17 
17 
17 
17 
17 
17 
17 
17 
17 

i 

17 

17 
17 
17 

17 

17 
17 
17 
17 
D 

10-31629 
.31684 
.31738 
•31793 
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5< 
5] 

5^ 
5' 
5; 

il 

54 

il 

il 

55 

55 
55 

55 
55 
55 
55 
55 
55 

11 

55 

| 

55 
55 

55 
56 
55 
55 
56 
55 
56 
56 
55 
56 
56 
56 
56 
56 
56 
56 
56 
56 
56 
56 
56 

0 

1 

2 
3 
4 

6 
7 
8 

9 
10 
20 
30 
40 
50 

6 
7 
8 

9 
10 
20 
30 
40 
50 

6 
7 
8 
9 
10 
20 
30 
40 
50 

6 
7 
8 
9 
10 
20 
30 
40 
50 

1 
6  1 
7  2 
8  2 
9  2 
10  3 
20  6 
30  9 
4012 
5015 

fl 

5 
6 
7 
8 
9 
18 
28 
37 
47 

r> 

6 
7 
8 

9 
18 

27 
37 
46 

5 

5 
6 
7 
8 

9 
18 

27 
36 
45 

5 

5 
6 
7 
8 
8 
17 
26 
35 
44 

7 
8 
9 
10 

20 
30 
40 
50 

8 
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•  1 
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1 

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•  4 
3_ 

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1 

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1 

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r 

21 
81 

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1 
1 
2 
2 
2 
2 
5 
8 
11 
14 

56 

5-6 
6-5 
7-4 
8-4 

9-3 
18-  6 
28-0 
37-3 
46-6 

55 

5-5 
6-- 
7-i 

8-i 

9. 

is.; 

27-5 
36- 
45-8 

54 

5-4 
6-3 
7-2 
8-1 
9-0 
18-0 
27-0 
36-0 
45-0 

53 

5-3 
6-2 

7-5 
8.8 

17-6 
26-5 
35-3 
44-1 

"i 

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':? 
:! 

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7  17 

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.0  2-0 

000 

5 
6 
7 
8 

9 
10 

11 
12 
13 
14 
15 
16 
17 
18 
ii 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 

33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

9-81911 
•8192! 
.81947 
.81965 
-81983 
9-82001 
-82019 
•  82037 
•82055 
•82073 

10-28680 
.28733 
.28786 
.28839 
.  28892 
10-28945 
.28998 
.29051 
.29104 
•29157 

9-82982 
•8300C 
.83017 
83035 
_.  83053 
9-83070 
.83088 
.83106 
.83123 
.-83141 
9-83159 
.83176 
•83194 
•83211 
-83229 

10-31902 
•31956 
.32011 
.32066 

10-32175 
•32230 
.32284 
.32339 
•32394 

5 
6 
7 
8 
9 
10 
11 

13 
14 
15 
16 

17 
18 
19 

9-82091 
•82109 
.82127 
•82145 
•82163 

10-29215 
•29263 
•29316 
•29370 
-29423 

10  -  32449 
.32504 
•32558 
•32613 
-32668 

9-82181 
-82199 
•82217 
•82235 
-82252 

10-29476 
.29529 
.29583 
.29636 
•29689 

9-83247 
•83264 
•83282 
.83299 
.83317 

10-32723 
.32778 
.32833 
.32888 
-32944 

30 

21 

22 
23 
24 

9-82270 
•  82288 
•82306 
-82324 
-82342 

10-29743 
•29796 
.29850 
.29903 
•29957 

9.83335 

•83352 
.83370 
.83387 
•83405 

10-32999 
•33054 
.33109 
.33164 
-33220 

25 
26 
27 
28 
29 

9-82360 
-82378 
-82391) 
•  8241J! 
-82431 

10-30010 
.30064 
.30117 
•30171 
.30225 

9  •  83422 
.83440 
.83458 
.83475 
.83493 

10-33275 
.33330 
.33385 
.33441 
.33496 

30 

31 
32 
33 
34 

9  -  82449 
-82467 
-82485 
•  82503 

-82520 

10-30278 
•30332 
.30386 
.30440 

.30493 

9-83510 
.83528 
.83545 
•83563 

-83580 

10.33552 
•33607 
•33663 
.33718 
-33774 

35 
36 
37 
38 
-39. 
40 
41 
42 
43 
.44 
45 
46 
47 
48 
49 

9-82538 
•  82556 
•82574 
•82592 
.82609 

18 

if 

18 
18 
17 
18 
17 

18 
17 
18 
17 
18 
17 
18 
17 
18 
•^ 

10-30547 
.30601 
.30655 
.30709 
.30763 

9-83598 
•83615 
•83633 
•83650 
.83668 

10-33829 
.33885 
•33941 
-33996 
-34052 

9-82627 
-82645 
•82663 
-82681 
-82698 

10-30817 
.30871 
•30925 
•30979 
•31033 

9-83685 
•83703 
.83720 
•83737 
•83755 

10-34108 
•34164 
•34220 
•34275 
•3433! 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 

9-82716 
-82734 
-82752 
-82769 
-82787 

10-31087 
•31141 
.31195 
.3124? 
.31303 

9-83772 
•83790 
•83807 
•83825 
•83842 

10-34387 
-  34443 
-34495 
.34555 
.34611 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 

9-82805 
-82823 
-82849 
-82858 
-82876 

10-31358 

•31412 
.31466 
•31521 
.31575 

9-83859 
.83877 
•83894 
•83912 
•83929 

10-34667 
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.34780 
.34836 
.34892 

-6  2-5 

!§  5J 
-7  8-5 
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.6  14-1 

go 

9-82894 
Lg.Vers. 

10-31629 

9-83946 

10-34948 

60 

Log.  Exs, 

D 

Lg.Vers 

Log,  Exs, 

' 

P,  P, 

LOG.   VERSED  SINES  AND  EXTERNAL  SECANTS 

73°  73° 


201 


1 

Lg.Vers 

It 

Log.Exs. 

D 

Lg.Vers 

D 

Log.  Exs. 

D 

' 

P.  P. 

0 

1 

9-83946 
.83964 

10-34948 
.35005 

56 

9-84980 
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17 

10-38387 
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58 

0 

1 

6 

61 

6-1 

60 

6-0 

2 
8 

.83981 
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17 
17 

•35061 
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56 

.85014 
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17 

1  7 

.38504 
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58 

2 
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7 
8 

7. 
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4 

9 

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9-84033 

12 

10-35230 

bb 

9-85066 

IV 

10-38679 

b8 

5 

10 

10- 

10-1 

6 

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8 

6 

20 

20. 

20-1 

7 

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17 

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56 

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17 

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58 

7 

30 

30-  b 

30-2 

8 

9 

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.84103 

17 

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57 

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17 

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59 
58 

8 
9 

40 
50 

40-  b 
50-8 

40  3 
50-4 

K 

11 

9-84120 
•84137 

IV 

iz 

10-35513 
.35569 

bb 
56 

9-85151 
.85168 

IV 
17 

10-38973 
•39031 

fa9 
58 

10 

11 

60 

59 

12 
13 
14 

.84155 
.84172 
.84189 

17 
17 

.35626 
.35685 
.35739 

57 
'56 

•85185 
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-85219 

17 
17 

.39090 
.39149 
•39208 

59 
59 
58 

12 
13 
14 

7 
8 

p 

7-0 
8-0 

90 

6-9 
7-9 

15 
16 
17 
18 
19 

9-84207 
.  84224 
.  84241 
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.84276 

17 

10-35796 
.35853 
.35910 
.35967 
.36023 

bV 
56 
57 

il 

9-85236 
-85253 
-85270 
-85287 
.85304 

IV 
17 
17 
17 
17 

10-39267 
•39326 
•39385 
•39444 
-3&5G3 

59 
59 
59 
59 
59 

15 
16 
17 
18 
19 

10 
20 
30 
40 
50 

10-0 
20-0 
30-0 
40-0 
50-0 

9-9 
19-8 
29-7 
39-6 
49-6 

20 

21 

9-84293 
.84310 

IV 
17 

10-36080 
•36137 

bV 
57 

9-85321 
•85338 

IV 
17 

10.3S562 
-89621 

59 
59 

20 

21 

59  58 

22 

.  84328 

|Z 

.36194 

57 

•85355 

1  7 

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M 

6 

b-9 

5-8 

23 

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28 

7 

6.9 

6-8 

24 

.84362 

•3630J 

.85389 

.39799 

24 

8 

V-8 

7-8 

25 
26 
27 
28 
29 

9-84380 
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.84414 
.  84431 
.  84449 

IV 
17 
17 

10-36366 
•36423 
•36480 
•36537 
•36594 

bV 
57 
57 
57 
57 

9  -  85405 
•8542J 
-8543< 
-85456 
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16 
17 
17 
17 
17 

10-39859 
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•39977 
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59 
59 
59 
59 
59 

25 
26 
27 
28 

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10 
20 
30 
40 

en 

8-8 
9-8 
19-6 
29-5 
39-3 

8-8 
9-7 
19-5 
29-2 
39-0 

30 

9-84466 

17 

10-36652 

bV 

9-85490 

IV 

10-40156 

59 

80 

81 
32 
33 
34 

•84483 
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-84535 

II 
II 

•36709 
•36766 
•36824 
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57 
57 
57 

•85507 
.85524 
•85541 
.85558 

II 

17 
17 

•40216 
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-4C335 
-40395 

60 
59 
59 
60 

31 
32 
33 
34 

6 
7 
8 

58 
5.8 

6-7 
7-7 

57 
5-7 
6-7 
7-6 

35 

9-84552 

IV 

1  7 

10-36938 

5Z 

9-85575 

IV 

10-40454 

59 

35 

9 

8-7 

8-6 

36 

.84569 

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57 

•85592 

i 

•40514 

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36 

10 

9-b 

9-6 

37 
38 
39 

.84586 
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17 
17 
17 

•37054 
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57 
57 
57 

K7 

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17 
17 
17 

•40574 
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b9 
60 
60 
60 

37 
38 

40~ 

20 
30 
40 
50 

19-3 
29-0 
38-6 

48-3 

19-1 
28-7 
38-3 
47-9 

40 

9-84838 

10-37226 

9-85659 

10-40754 

41 
42 
43 
44 

•84655 
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17 
17 
17 

•37284 
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58 
57 
58 

•85676 
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•85710 
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17 

11 

.40814 
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60 
60 
60 
60 

41 
42 
43 
44 

6 
7 

57 

5-7 
6-6 
7  B 

56 

5-6 
66 
7  5 

45 
46 

9-84724 
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IV 
17 

10-37515 
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b8 
57 

9-85743 
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IV 

il 

10-41054 
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60 
60 

45 
46 

9 
10 

Li 
9.5 

8-5 
9-4 

47 

•84758 

1  7 

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CO 

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1  7 

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60 

47 

20 

19-0 

18-8 

48 

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c 

48 

30 

28-5 

28-2 

49 

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17 

1  7 

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58 

CO 

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16 

1  7 

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10-41355 

60 
fin 

49 

40  38  -  C 
50147.5 

37-6 
47-1 

50 

9-84809 

10-37805 

9-85827 

50 

bi 
52 
53 
54 

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•  84844 
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1-7 
17 
17 

•37863 
•37921 
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58 
58 
58 

•85844 
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•85878 
.85895 

il 

17 

•41416 
•41476 
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60 
60 
60 

51 

52 
53 
54 

17  17  16 

6  1-7  1-7  1-6 
7  2-0  2-0  1-9 

55 

9-84895 

17 

10-38095 

58 

KO 

9-85911 

Ib 
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10-41658 

60 

55 

9  2-6  2-5  25 

bb 

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1 

56 

10  29  28  27 

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58 
59 
60 

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17 
17 
17 

•3821P 
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58 
58 

58 

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-85979 

s§ 

17 
16 

•41779 
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60 
60 
61 
61 

57 
58 
59 

20  5-8  5-6  5-5 
30  8-7  85  8-2 
4011-6  11  3  11-0 
FP  14.6  14-  T  '3-7 

9-84980 

10-3S387 

0  85995 

10-41962 

60 

Lg.  Vers. 

^ 

Log.Exs. 

5 

Lg.Vers. 

D 

Log.  Exs. 

Z> 

' 

P.  P. 

202    LOG.   VERSED  SINES   AND  EXTERNAL  SECANTS 
74°  75° 


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20 

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10-64371 
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20 

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24 
25 
26 
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9-90266 
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10-59991 
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55 
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LOG.   VERSED  SINES   AND  EXTERNAL   SECANTS    205 
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25 
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35 
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10-70859 
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10-76581 
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35 
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30 

31 
32 
33 
34- 
35 
36 
37 
38 
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40 
41 
42 
43 
44 

9-95693 
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10-98197 
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9-96519 
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11-07865 
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1-3 

1-5 
3-0 

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7-5 

0 

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13 

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40 

41 
42 
43 
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9-95763 
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14 
14 
13 
14 
14 

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14 
14 

11 

14 
14 
13 
14 

11 

V. 

10-98938 
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9-96588 
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11-08759 
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45 
46 
47 
48 
49 

9-95835 
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10-99689 
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10-99993 
11-00145 
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9-96656 
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11-09669 
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.  10409 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

9-95901 
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11-00451 
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Ib3 
154 
154 
155 
155 
155 
156 
156 
157 
157 
158 
D 

9-96724 
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11-10595 
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50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

9-95976 
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11-01225 
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01692 
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9-96792 
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11-11539 
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55 
56 
57 
58 
59 

§Q 

9-96039 

11-0201C 

9-96859 

11-12501 

60 

Lg,  Vers 

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Log.Exs. 

Lg.Vers 

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Log.Exs, 

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P.  P, 

208    LOG.  VERSED  SINES  AND   EXTERNAL  SECANTS 

86°  87° 


Lg.  Vers 

D 

Log.  Exs. 

D 

Lg.  Vers. 

D 

Log.  Exs. 

D 

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P.  P. 

0 

1 

2 
3 
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9-96859 
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13 

13 
13 

11-12501 
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19 
19 
19 
19 

9-97665 
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13 
13 
13 
13 

11.25785 
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255 
256 
257 
259 

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2 

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6 
7 
8 
9 

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33-3 
37-5 

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24-0 
28.0 
32-0 
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5 
6 
7 
8 

9 

9-96927 
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11 
11 
13 

11.13482 
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19 
19 
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20 
20 

9.97732 
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lei 
13 
13 
13 
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11.27074 
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261 
262 
263 
265 
266 

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10 
20 
30 
40 
50 

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10 

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11-14482 

9-97798 

11-28398 

267 

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230     220 

11 
12 
13 
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13 

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203 
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13 
13 
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272 
274 

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12 
13 
14 

6 
7 
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9 

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15 
16 
17 
18 
19 

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13 
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11.15502 
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208 
208 
209 

9-97864 
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13 
13 
13 

13 

11-29760 
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277 
278 
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16 
17 
18 
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10 
20 
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20 

21 
22 
23 
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1-16544 
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210 
211 
212 
213 

214 

9.97931 
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13 
13 
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11.31162 
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287 
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220 

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9-98063 
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13 
13 
13 
13 
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11-34095 
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301 
303 
305 
307 
309 

30 
31 
32 
33 
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6 
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1-20947 

224 
225 
227 
227 
228 

230 
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9.98129 
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13 
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13 
13 
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313 
315 
318 
320 
322 

35 
36 
37 
38 

39 
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43 
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45 
46 
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13 
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237 
239 
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9.98260 
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13 
13 
13 
13 
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11-38866 
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666 
335 
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340 
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13 
13 
13 

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243 
245 
246 

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13 
13 
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351 
353 
356 

51 
52 
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54 

6 
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3     13 

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9-98392 

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251 
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13 
13 

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367 
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57 
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59 

20 
30 
40 

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60 

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13 

11-25785 

264 

9-98457 

13 

11.44175 

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D 

Log.  Exs. 

D 

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D 

Log.  Exs, 

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P.P. 

LOG.   VERSED   SINES  AND  EXTERNAL  SECANTS 

88°  89° 


209 


Lg.Vers 

It 

Log.Exs. 

D 

Lg.Vers. 

D 

Log.  Exs 

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0 

1 

2 
3 
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13 
13 
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11-44175 
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37 
379 
382 
386 

9.99235 
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13 
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4 

5 
6 
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9-98522 
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13 
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13 
13 
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11.46088 
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389 
392 
395 
399 
402 

9-99299 
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11-7889 
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840 
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7 
8 
9 

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14 

9-98588 
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13 
13 
13 
13 
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11.48083 
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4Ub 
409 
413 
417 
420 

9-99363 
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12 
13 

12 
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11.83095 
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872 
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10 

11 
12 
13 
14 

15 
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17 
18 
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9-98693 
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13 
13 
13 
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11-50168 
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9-99428 
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13 
12 
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11-87735 
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967 
989 
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15 
16 
17 
18 

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20 

21 
22 
23 
24 

9-98718 
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13 
13 
13 
13 
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11-52351 
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445 
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454 
458 
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9-99491 
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13 
13 
12 
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11-92914 
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1112 
1140 
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1203 

20 

21 
22 
23 
24 

25 
26 
27 

28 

29 

9-98783 
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13 
13 
13 
13 
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11-54643 
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477 
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9-99555 
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12 
13 
12 
13 
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11.98777 
12.00048 
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1236 
1271 
1309 
1349 
1391 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

9-98848 
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13 
13 
13 
13 
13 

11-57056 
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492 
498 
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509 
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9-99619 
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13 
12 
13 

12-05535 
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31 
32 
33 
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37 
38 
39 

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12 
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12-13517 
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1785 
1861 
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35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

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13 
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1.62300 
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12-23271 
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2131 
2240 
2361 
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41 
42 
43 
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7  1.6  1. 
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45 
46 
47 
48 
49 

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13 
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12 
13 
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13 

12-35828 
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46 
47 
48 
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10  2-2  2. 
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50 

51 
52 
53 
54 

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13 
13 
13 
13 
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1.68227 
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628 
638 
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656 
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9-99873 
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12 
13 
12 
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2-53501 
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4152 
4588 
5127 
5812 
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50 

51 
52 
53 
54 

12 

6  1-2 
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8  1.6 

55 
56 
57 
58 
59 

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13 
13 

12 
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1-71509 
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9-99937 
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18 
12 
12 
12 
13 

2-83667 
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55 
56 
57 
58 
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20  4-1 
30  6.2 
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30 

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Infinity 

60 

' 

Lg.Vers. 

J> 

Log.Exs, 

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Lg.Vers.  | 

X> 

Log.Exs, 

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7 

P,  P, 

210 


THE   RAILROAD   TAPER 


FUNCTIONS  OF  A  ONE-DEGREE  CURVE 

The  Long  Chords,  Mid-Ordinates,  Externals,  and  Tangent 
Distances  of  this  table  are  for  a  curve  of  5730  feet  radius.  To 
find  the  corresponding  functions  of  any  other  curve  divide  the 
tabular  values  by  the  degree  of  curve. 

For  metric  curves  having  a  20-meter  chord,  this  is  a  table  of 
the  functions  of  a  two-tenth  degree  curve.  Multiply  the  values  in 
the  table  by  0.2  to  obtain  the  functions,  in  meters,  of  a  one-degree 
metric  curve. 


0° 

1° 

/ 

L.  0.       M.            E.           T. 

L.  C.        M.           E.           T. 

0 

0.00     0.000      0.000        0.00 

100.00     0.',M8     0.218       50  00 

0 

2 

3.33     0.000      0.000        1.67 

103.33     0.233     0.233       51.67 

2 

4 

6.6?     0  001      0.001        3.33 

106.66     0.2J8     0.248       53.33 

4 

6 

10.00     0.002      0.002        5.00 

310.00     0.264     0.264       55.00 

6 

8 

13.33     0.004      0.004        6.67 

113.33     0.280     0  280       56.67 

8 

10 

16.67     0.006      0.006        8.33 

116.66     0.297     0.297       58.33 

10 

12 

20.00     0.009      0.009      10.00 

120  00     0.314     0.314       60.00 

12 

14 

23.33     0.012      0.012      11.67 

123.33     0.332     0.332       61.67 

14 

16 

26.67     0.015      0.015      13.33 

126.66     0.350     0.350       63  33 

16 

18 

30.00     0.019      0.019      15.00 

130.00     0.3G8     0.368       65.00 

18 

20 

33.33     0.024      0.024      16.67 

133.33     0.388     0.388      66  67 

20 

22 

36.67     0.029      0.029.     18.33 

136.66     0.407     0.407       68.33 

22 

24 

40.00     0.035      0.035      20.00 

140.00     0.427     0.427       70  00 

24 

26 

43.33     0.041      0.041      21.67 

143.33     0.448     0.448       71.67 

26 

28 

4667     0.048      0.048      23.33 

146.66     0.469     0.469       73.33 

28 

30 

50.00     0.054      0.054      25.00 

15000     0.491      0.491        75.00 

30 

32 

53  33     0  062      O.OC2      26.67 

153.33     0.513     0.513       76.67 

32 

34 

5667     0070      0070      28.33 

156.66     0.536     0.536       78.33 

34 

36 

6000     0.079      0.079      30.00 

160.00     0.559     0.559       80.00 

36 

38 

63.33     0.088      0.088      31.67 

163.33     0.582     0.582       81.07 

38 

40 

66.67     0.097      0.097      3333 

166  66     0.606     6.606       83.33 

40 

42 

70.00     0.107      0.107      35.00 

J70.00     0.630     0.630       85.00 

42 

44 

73.33     0.117      0.117      36.  G7 

173  33     0.655     0.655       86.67 

44 

46 

76.6?     0128      0.128      38.33 

176.66     0.681     0.681       88  33 

46 

48 

80.00     0.140      0.140      40.00 

180.00     0.706     0.706       90.00 

48 

50 

83.33     0.151       0.151      41.67 

18333     0.733     0.733       91.07 

50 

52 

86  67     0.164      0.164      43.33 

186.66     0.760     0.760       93.33 

52 

54 

90  00     0  176      0.176      45.00 

190.00     0.788     0788       95.00 

54 

56 

9333     0.190      0.190      46.67 

193.33     0.815     0.815       96.67 

56 

58 

96.67     0.204      0.204      4833 

196.66     0.844     0.844       98.33 

58 

60 

100.00     0.218      0.218      5000 

199.98     0.873     0873     100.00 

60 

FUNCTIONS  OF  A  ONE-DEGREE  CURVE       211 


Jjo 

3° 

9 

L.  C.         M.          E.           T. 

L.  C.         M.          E.           T. 

/ 

o 

199.98      0.873      0.873    100.00 

299.96      1.964      a.  964    150.07 

0 

2 

203.31      0.902      0.902    101.67 

303.29      2.008      2.009    151  74 

2 

4 

206.64      0.932      0.932    103.34 

306.62      2.053      2.054    153.41 

4 

6 

209.97      0.9C2      0.962    105.01 

309.95      2.098      2.099    155  08 

6 

8 

213.31      0.993      0.993    106.68 

313.29      2.143      2.144     156.75 

8 

10 

216.64         .024      1.024    108.35 

316.62      2.188      2.189    158.42 

10 

12 

219.97         .056      1.056    110.02 

319.95      2.235      2.236     160.09 

12 

14 

223.30        .088      1.088    111.69 

323.28      2.282      2.283    161.76 

14 

16 

226.64         .121       1.121     113.36 

326.62      2.329      2.330     163.43 

16 

18 

229.97         .154      1.154     115.02 

329.95      2.376      2.377     165.09 

18 

20 

233.30        .188      1.188    116.69 

333.28      ?.424      2.425    166.76 

20 

22 

236.63         .222      1.222     118.36 

336.61      2.473      2.474     168.43 

22 

24 

239.97         .256      1.256    120.03 

339.95      2.523      2.523     170.10 

24 

26 

243.30        .292      1.292    121.70 

343.28      2.572      2.573     171.77 

26 

28 

246.63         .328      1.328    123.37 

346.61      2.622      2.623     17344 

28 

30 

249.96         .364      1.364     125.03 

349.94      2.672      2.G73     175.10 

30 

32 

253.29        .399      1.399    126.70 

353.27      2.724      2.725    176.72 

32 

34 

256.62         .437      1.437    128.37 

356.60      2.776      2  777    178.39 

34 

36 

259.96        .475      1.475    130.04 

359.94      2.828      2.829     180.06 

36 

38 

263.29        .513      1.513    131.71 

363.27      2.880      2.881     181.73 

38 

40 

266.62         .552      1.552    133.38 

366.60      2.933      2.934     183.40 

40 

42 

269.96         .592      1.592    135.05 

369.94      2.987      2.988     185.07 

42 

44 

273.29         .632      1.632     136.72 

373.27      3.042      3  043     186.74 

44 

46 

276.62         .672      1.672    138.38 

376.60      3.096      3  097    188.40 

46 

43 

279.96         .712      1.712    140.05 

379.94      3.151      3.152    190.07 

48 

50 

283.29        .752      1.752     141.72 

383.27      3.206      3.207     191.74 

50 

52 

286.62        .794      1.794    143.39 

386.60      3.263      3.264     193.41 

52 

54 

289.96         .836      1.836    145.06 

389.94      3.320      3.321     195.08 

54 

56 

293.29         .878      1.878     146.73 

393.27      3.377      3  378     196.75 

56 

58 

296.62         .921       1.921     148.40 

396.60      3.434      3.435     198.42 

58 

60 

299.96        .964      1.964    150.07 

399.94      3.491       3.492    2uO  09 

60 

4° 

6° 

L.  C.         M.           E.          T. 

L.  C          M.          E.          T. 

8 

0 

399.94      3.491       3.492    200.09 

499.88      5.454      5.459    250.17 

0 

2 

403.27      3.550      3.551     201.76 

503.21      5.&27      5.o33    251.84 

2 

4 

406.00      3.609      3.610    203.43 

506.54      5.601      5.607    253.51 

4 

6 

409.93      3.668      3.670    205.10 

509.87      5.675      5681    255.18 

6 

8 

413.26      3.727      3.730    206.77 

513.20      5.749      5.755    256.85 

8 

10 

416.59      3.787      3.790    208.44 

51653      5.823      5.829    258.52 

10 

12 

419.92      3.848      3.851    210.11 

519.86      5.899      5.905    260.20 

12 

14 

423.26      3.910      3.913    211.77 

523.19      5.975      5  981     261.86 

14 

16 

426.59      3.972      3.975    213.45 

526.52      6.052      6.058    263.54 

16 

18 

429.92      4.034      4.037    215.11 

529.85      6.129      6.135    265.20 

18 

20 

433.25      4.096      4.099    216.78 

533.18      6.206      6.212    266.87 

20 

22 

436.58      4.160      4.163    218.45 

536.51      6.284      6.290    268.54 

22 

24 

439.91      4.224      4.227    220.12 

539.84      6.362      6.369    270.21 

24 

26 

443.24      4.288      4.291     221.79 

543.17      6.441      6.448    971.88 

26 

28 

446.58      4.353      4.356    223.46 

546.50      6.520      6.527    273.54 

28 

30 

449.91      4.418      4.421     225.13 

549.83      6.599      6.606    275.21 

30 

32 

453.24      4.484      4.487    226.80 

553  17      6.680      6.687    276.88 

32 

34 

456.57      4.550      4.554    228.47 

556.50      6.761       6.768    278.55 

34 

.36 

45990      4.617      4.621     230.14 

559.83      6.842      6.849    280.23 

36 

38 

463.23      4.684      4.G88    231.81 

563.16      6.923      6.931     281.90 

38 

40 

466.56      4.751      4.755    23348 

566.49      7.005      7013    283.57 

40 

42 

469.89      4.820      4.824    235.15 

569.82      7.088      7.096    285.24 

42 

44 

473.23      4.889      4.6Q3    236.82 

573.15      7  171      7.180    28691 

44 

46 

476  56      4.958      4  962    238.48 

576.48      7  255      7.264    288.59 

46 

48 

479.89      5.027      5.031     240.15 

579.81      7339      7.348    290.26 

48 

50 

483.22      5.096      5  100    241.82 

583.14      7423      7.432    291.93 

50 

52 

486.55      5.167      5.171     243.49 

586.47      7.508      7.517    293.60 

52 

54 

489  88      5.238      5.243    245.16 

589.80      7.593      7.603    295.27 

54 

56 

493.21       5.310      5.315    24683 

593.13      7.678      7  689    296  95 

56 

58 

496  54      5.382      5.387    248.50 

596.46      7.764      7  775    298.62 

58 

60 

499.88      5  454      5.459    250.17 

599.80      7.850      7  861     300.30 

60 

212       FUNCTIONS  OF  A  ONE-DEGREE  CURVE 


6° 

7° 

/ 

L.  C.         M.           E.         T. 

L.  C.         M.           E.           T. 

/ 

0 

599.80      7.850      7.861    300.30 

699.60     10.69     10.71     350.44 

0 

2 

603.13      7.940      7.951    301.97 

702.93      10.79     10.81     352.11 

2 

4 

606  46      8.030      8.041    303.64 

706.26     10.90     10.92     353.79 

4 

6 

609.78      8.120      8.131    305.31 

709.58     11.00     11.02     355.46 

6 

8 

613.11      8.210      8.221    306.98 

712.91      11.11     11.13     357.13 

8 

10 

616.44      8.300      8.311    308.65 

716.24     11.21     11.23     35881 

10 

12 

619.76      8.390      8.401    310.32 

719.56     11.31     11.33     360.48 

12 

14 

623.09      8.480      8.491    311.99 

722.89     11.42     11.44     362.15 

14 

16 

626.42      8.570      8.581    313.66 

726.21      11.52     11.54     36383 

16 

13 

629.74      8.660      8.671    315.33 

729.53     11.63     11.65     365.50 

18 

20 

633.07      8.750      8.761    317.00 

732.86     11.73     11.75     367.17 

20 

22 

636.40      8.844      8.856    318.67 

736.19      11.84     11.86     368.85 

22 

24 

639.72      8.939      8.951    320.34 

739.51      11.95     11.97     370.52 

24 

26 

643.05      9.033      9.046    322.01 

742.84      12.06     1208     37219 

26 

28 

646.38      9.128      9.141    323.68 

746.17     12.17     12.19     373.80 

28 

30 

649.70      9.222      9.236    325  35 

749.49     12.27     12.30     375.54 

30 

32 

653.03      9.317      9.331    327.02 

752.82     12.38     12.41     377.22 

32 

34 

656.36      9.411      9.426    328.69 

756.15     12.49     12  52     378.  H9 

34 

36 

659.69      9.506      9.521    330.37 

759.47     12.60     12.63     380  57 

36 

38 

663.02      9.600      9.616    332  04 

762.80     12.71     12.74     382.24 

38 

40 

666.34      9.695      9.712    333.71 

766.13     12.82     12.85     383.92 

40 

42 

669.67      9  794      9.812    335.38 

769.45     12.93     12.96     385.60 

42 

44 

673.00      9.894      9.912    337.05 

772.78     13.04     13.08     387.27 

44 

46 

676.32      9.993    10.01      338.78 

776.11      13.15     13.19     388.95 

46 

48 

679.65    10.09      10.11      340.40 

779.43     13.26     13.31     390.62 

48 

50 

68298     10.19      10.21      342.07 

782.76     13.37     13.42     392.30 

50 

52 

686.30    10.29      10.31      343.74 

786.09     13.48     13.53     393.98 

52 

54 

68963    10.39      10.41      345.41 

789.41      13.59     13.65     395.65 

54 

56 

692.96    10.49      10.51      347.08 

792.74     13-.  70     13.76     397.33 

56 

58 

696.28    10.59      10.61      348.76 

796.07     13.81     13.88     399.01 

58 

60 

699.60    10.69      10.71      350.44 

799.40     13.96     13.99     400.70 

60 

8° 

9° 

L.  C.        M.           E.           T. 

L.  C.        M.          E.           T. 

0 

799.40     13.96      13.99     400.70 

899.10     r.C6     17.71     450.95 

0 

2 

802.72     14.07       14.10     402.37 

902.42     17.79     17.84    452.63 

2 

4 

806.04     14.19      14.22     40405 

905.74     17.92     17.98    454.31 

4 

6 

809.37     14.31       14.34     405.72 

909.07     18.06     18.11    455.98 

6 

8 

812.69     1443      14.46     407.39 

•912.39     18.19     18.25    457.66 

8 

10 

816.01     14.55      14.58     409.06 

915.71      18.32     18.38    459.34 

10 

12 

819.34     14.66      14.70     410.74 

919.04     18.46     18.52    461.02 

12 

14 

822.66     14.78      14.82     412.41 

922.36     18.59     18.65    462.70 

14 

16 

825.98     14.90      14.94     414.08 

925.68     18.72     18.79    464.37 

16 

18 

829.31     15.02      15.06     415.75 

929.01     18.86     18.92    466.05 

18 

20 

832.63     15.14      15.18     417.43 

932.33     18.99     19.06    467.73 

20 

22 

835.95     15.26      15.30     419.10 

935.65     19.12     19.19    469.41 

22 

24 

839.28     15.38      15.43     420.77 

938.98     19.26     19.33    471.08 

24 

26 

842.60     15.51       15.55     422.45 

942.30     19.40     19.47    472.76 

26 

28 

845.92     15.63      15.68     424.12 

945.62     19.54     19.61    474.43 

28 

30 

849.25     15.75       15.80     425.79 

948.95     19.68     19.75    476.10 

30 

32 

852.57     15.88      15.93     427.47 

952.27     19.82     19.89    477.78 

32 

34 

855.89     16.00      16.05     429.15 

955.59     19.96     20.03    479.46 

34 

36 

859.22     16.12      16.18     430.82 

958.92     20.10     20.17    481.14 

36 

38 

862.54     16.25      16.30     432.50 

962.24     20.24     20.31    482.83 

38  ' 

40 

865.86     16.38      16.43     434.18 

965.56     20.38     20.45    484.51 

40 

42 

869.19     16.50      16.55     435.86 

968.89     20.52     20.59    486.19 

42 

44 

872.51     16.63      16.68     437.54 

972.21      20.66     20.74    487.87 

44 

46 

875.83     16.76      16.81     439.21 

975.53     20.80     20.88    489.56 

46 

48 

879.16     16.89      16.94     440.89 

978.86     20.94     21.03    491.24 

48 

50 

882.48     17.02      17.07     442.57 

982.18     21.09     21.17    492.92 

50 

52 

885.80     17.14      17.19     444.25 

985.50     21.23     21.31     494.60 

52 

54 

889.13     17.27      17.32     445.93 

988.83     21.37     21.46    496.28 

54 

56 

892.45     17.40      17.45     447.60 

992.15     21.51     21.60    497.96 

56 

58 

895.77     17.53       17.58     449.28 

995.47     21.65     21.75    499.65 

58 

60 

899.10     17.66       17.71      450.95 

998.80     21.80     21.89    501.32 

60 

FUNCTIONS  OF  A  ONE-DEGREE   CURVE      213 


10° 

11° 

' 

L  C.        M.          E.         T. 

L.  C.         M.           E.          T. 

/ 

0 

998.8      21.80      21.89    501.32 

1098.4     26.38      26.50     551.74 

0 

2 

1002.1      21.94      22.03    503.00 

1101.7     26.54      26.86     553.42 

2 

4 

1005.4      22.09      82.18    504.68 

1105.0     26.70      26.83     555.10 

4 

6 

1008.8      22.24      22.33    506.36 

1108  3     26.86      26.99     556.78 

6 

8 

1012.1      22.39      22.48    508.04 

1111.7     27.02      27.16     558.46 

8 

10 

1015.4      22.54      22.63    509.72 

1115.0     27.19      27.32     560.14 

10 

12 

1018.7      22.68      22.78    511.40 

1118.3     27.35      27.48     561.82 

12 

14 

1022  0      32.83      22.93    513.08 

1121.6     27.51      27.65     563.50 

14 

16 

1025.4      22.98      23.08    514.76 

1124.9     27.67      27.81     565.18 

18 

18 

1028.7      23.13      23.23    516.44 

1128.2     27.83      27.98     566.86 

18 

20 

1032.0      23.28      23.38    518  12 

1131  6     28.00      28.14     568.54 

20 

22 

1035.3      23.43      23.53    519.80 

11349     26.17      28.30     570.22 

22 

24 

1038.6      23.58      23  68    521.48 

1138.2     28.34      28.47     571.90 

24 

26 

1042.0      23.73      23.84    523.16 

1141.5     28.50      2864     573.58 

26 

28 

1045.3      23.88      23.99    524.85 

1144.8     28.67      28.81     575  27 

28 

30 

1048.6      24.04      24.14    526.53 

1148.1     2884      28.98     576.95 

30 

32 

1051.9      24.19      24.30    528.21 

1151.5     29.00      29.14     578.63 

32 

34 

1055.2      24.34      24.45    529.89 

1154.8     29.17      29.31      580.32 

34 

36 

1058.6      24.49.      24.60    531.57 

1158.1      29.34      29.48     582.00 

36 

.38 

1061.9      24.64      24.76    53325 

1161.4     29.50      29.05     583.69 

38 

40 

1065.2      24.80      24.91     534.93 

1164.7     29.67      29.82     585.37 

40 

42 

1068.5      24.95      25.06    536.65 

1168.0     29.84      29.99     587.05 

42 

44 

1071.8      25.11      25.22    53829 

1171  4     30.01      30.17     588.74 

44 

46 

1075.2      25.27      25.38    539.97 

1174.7     30.18      30.34     590.42 

46 

48 

1078.5      25.43      25.54    541.65 

1178.0     30.35      30.52     592.11 

48 

50 

1081.8      25.59      25.70    54333 

1181.3     30.53      3069     593.79 

50 

52 

1085.1      25.74      25.86    545.01 

1184.6     30.70      30.86     595  47 

52 

54 

1088.4      25.90      26.02    546.69 

1187.9     30.87      31.04     597.16 

54 

56 

10U1.8      26.06      26.18    548.37 

1191.3     31.04      31.21     598.84 

56 

58 

1095.1      26.22      26.34    550  06 

1104.6     31.21      31.39     600.53 

58 

60 

1098.4      26.38      26.50    551.74 

1197.9     31.39      31.56     602.22 

60 

12° 

13° 

L.  C.         M.           E.          T. 

L.  C.         M.           E.          T. 

0 

1197.9      31.39      31.56    602.22 

1297.3     36.83      37.07     652.87 

0 

I 

1201.2      31.57      31.73    603.91 

1300.6     3".  02      37.26     654.56 

2 

4 

1204.5      31.74      31.91     605.60 

1303.9     3".  21      37.46     656.25 

4 

6 

1207.8      31.93      32.09    607.28 

1307.2     3".  40      37.65     657.93 

6 

8 

1211  1       32.09      32.27    608.97 

1310.5     3   .59      37.85     659.62 

8 

10 

1214.5      32.27      32.45    610.66 

1313.8     3-79      38.04     661.31 

10 

12 

1217.8      32.45      32.63    612.35 

1317.2     3".  98      38.23     663.  CO 

12 

14 

1221.1      32.62      32.81     614.04 

1*20.5     38.17      38.43     664.69 

14 

16 

1224.4      32.80      32.99    615.72 

1323.8     38.86      38.  C2     666.37 

16 

18 

1227.7      32.97      33.17    617.41 

1327.1     38.55      38.82     668.06 

18 

20 

1231.0      33.15      33.35    619.10 

1330.4     38.75      39.01     669.75 

20 

ss 

1231.3      33.33      33.53    6'.!0.79 

1333.7     38.95      39.  2C     671.44 

22 

24 

1237.7      33.51      33.72    622  48 

1337.0     39.15      39.40     673.13 

24 

26 

1241.0      33.69      33.90    624.16 

1340.3     39.35      39.60     674.81 

26 

28 

1244.3      33.87      34.09    6.  '5.  85 

1343.6     39.54      39.80     676.51 

28 

30 

1247.6      34.06      34.27     627.55 

1346.9     39.74      40.00     678.20 

30 

32 

1250.9      34.24      34  45    629.24 

1350.3     39.94      40.19     679.89 

32 

34 

1254.2      34.42      34  64    630.93 

1353.6     40.13      40.39     681.58 

34 

36 

1257.5      34,60      34  82    632.61 

1356.9     40.33      40  59     683.26 

36 

38 

1260.8      34.78      35.01     634.30 

1360.2     40.52      40.79     684.95 

38 

40 

1264.2      34  97      35.19    635.99 

1363,5     40.71      40.99     686.64 

40 

42 

1267.5      35.16      35.37    637.68 

1366.8     40.91       41.19     68833 

42 

44 

1270.8      36.34      35.56    639.37 

1370.1      41.11       41.40     690.02 

44 

I    46 

1274.1      35.53      35.75    641.05 

1373.4     41.31      41.60     691.70 

46 

48 

1277.4      35.71      85.94    642.74 

1376.7     41.51      41.81     693.39 

48 

50 

1280.7      35.90      36.13    644.43 

1380.0     41.71       42.01     695.08 

50 

52 

1284.0      36.09      36.31     646.12 

1383.4     41.91       4221      696.77 

52 

54 

1287.4      36.27      36.50    647.81 

1386.7     42.11      42.42     698.46 

54 

56 

1290.7      36.46      30.09    619.49 

1390.0     4231       42.62     700.14 

56 

58 

1294.0      36.64      36.88    651.18 

1393.3     42.51      42.83     701.83 

58 

60 

1297.3      36.83      3r.07    652.87 

1396.6     42.71       43.03     703.53 

60 

214       FUNCTIONS  OF  A  ONE-DEGREE  CURVE 


14° 

16° 

/ 

L.  C.        M.         E.          T. 

L.  C.        M.         E.          T. 

/ 

0 

1396.6    42.71    43.03    ?03.53 

1495.9    49.02    49.44    754.85 

0 

2 

1399.9    42.92    43.23    705.23 

1499.2    49.24    49.66    756.05 

2 

4 

1403.2    43.12    43.44    706.92 

1502.5    49.46    49.89    757.74 

4 

6 

1406.5    43.33    43.65    708.62 

1505.8    49.68    50.11     759.44 

6 

8 

1409.8    43.53    43.86    710.31 

1509.1    4990    50.34    761.13 

8 

10 

1413.1    43.74    44.07    712.01 

1512.4    50.12    50.56    762.83 

10 

12 

1416.5    43.94    44.28    713.71 

1515.7    50.34    50.78    764.53 

12 

14 

1419.8    44.15    44.49    715.40 

1519.0    50.56    51.01    766.22 

14 

16 

1423.1    44.35    44.70    717.10 

1522.3    5078    51.23    767.92 

16 

18 

1426.4    44.56    44.91    718.79 

1525.6    51.00    51.46    769.61 

18 

20 

1429.7    44.77    45.12    720.49 

1528.9    51.22    51.68    771.31 

20 

22 

1433.0    44  98    45.33    722.20 

1532.2    51.44    51.90    773.01 

22 

24 

1436.3    45.19    45.54    723.89 

1535.5    51.67    52.13    774.70 

24 

26 

1439.6    45.40    45.76    725.59 

1538.8    51.89    52.36    776.40 

26 

28 

1442.9    45.61     45.97    727.28 

1542.1     52.12    52.59    778.09 

28 

30 

1446  2    45.82    46.18    728.97 

1545.4    52.34    52.82    779.79 

30 

32 

1449.6    46.03    46.40    730.66 

1548.7    52.57    53.05    781.49 

32 

34 

1452.9    4^.24    46.61     732.35 

1552.0    52.79    53.28    783.19 

34 

36 

1456.2    46.45    46.82    734.05 

1555.3    53.02    53.51    784.89 

36 

38 

14M).5    46.66    47.04    785.74 

1558.6    53.24    53.74    786.59 

38 

40 

1462.8    46.87    47.25    737.43 

1561.9    53.47    53.97    788.29 

40 

42 

1460.1     47.08    47.46    739.12 

1585.2    53.69    54.20    789.99 

42 

44 

1469.4    47.30    47.68    740.81 

1568.5    53.92    54.44    791.69 

44 

46 

1472.7    47.51    47.90    742.51 

1571.8    54.15    54.67    793.39 

46 

48 

1476.0    47.73    48.12    744.20 

1575.1    54.38    54.91    795.09 

48 

50 

1479.3    47.94    48.34    745.89 

1578.4    54.61    55.14    796.79 

50 

52 

1482.7    48.16    48.56    747.58 

1581.7    54.84    55.37    798.49 

52 

54 

1486.0    48.37    48.78    749.27 

1585.0    55.07    55.61    800.19 

54 

56 

1489.3    48.59    49.00    750.97 

1588.3    55.30    55.84    801.89 

56 

58 

1492.6    48.80    49.22    752.66 

1591.6    5553    56.08    803.59 

58 

60 

1495.9    49.02    49.44    754.35 

1594.9    55.76    56.31     805.29 

60 

16° 

17° 

L.  C.        M.         E.         T. 

L.  C.        M.         E.          T. 

/ 

0 

1594.9    55.76    56.31    805.29 

1693.9    62.94    63.64    856.35 

0 

2 

1598.2    55.99    56.54    806.99 

1697.2    63.18    63.89    858.05 

2 

4 

1601.5    56.  S3    56.78    808.64 

1700.5    63.43    64.15    859.76 

4 

6 

1604.8    56.46    57.02    810.39 

1703.8    63.68    64.40    861.46 

6 

8 

1608.1     56.70    57.26    812.09 

1707.1    63.93    64.66    863.16 

8 

10 

1611.4    56.93    57.50    813.79 

1710.4    64.18    64.91    864.87 

10 

12 

1614.7    57.17    57.74    815.49 

1713.7    64.42    65.16    866.57 

12 

14 

1618.0    57.40    57.98    817.19 

1716.9    64.67    65.42    868.27 

14 

16 

1621.3    57.64    58.22    818.89 

1720.2    64.92    65.67    869.98 

16 

18 

1624.6    57.87    58.46    820.59 

1723.5    65.17    65.93    871.68 

18 

20 

1627.9    58.11    58.70    822.29 

1726.8    65.42    66.18    873.38 

20 

22 

1631.2    58.34    58.94    823.99 

1730.1    65.67    66.43    875.09 

22 

24 

1634.5    58.58    59  19    825.69 

1733.4    65.93    66.69    876.79 

24 

26 

1637.8    58.82    59.43    82739 

1736.7    66.18    66.95    878.49 

26 

28 

1641.1    59.06    59.68    829.09 

1740.0    66.44    67.21    880.20 

28 

30 

1644.4    59.30    59  92    83079 

1743.3    66.69    67.47    881.90 

30 

32 

1647.7    59.54    60.16    832.49 

1746.6    66.94    67.72    883.61 

32 

34 

1651.0    59.78    6041    834.20 

1749.9    67.20    67.98    885.32 

34 

36 

1654.3    60.02    60.65    835.90 

1753.2    67.45    68.24    887.02 

36 

38 

1657.6    60.26    60.90    837.61 

1756.5    67.71     68.50    888.73 

38 

40 

1660.9    60.50    61.14    839.31 

1759.8    67.96    68.76    890.44 

40 

42 

1664.2    60.74    61.39    841.01 

1763.1     68.21     69.03    892.15 

42 

44 

1667.5    60.99    61.64    842.72 

1766.3    68.47    69.29    893.86 

44 

46 

1670.8    61.23    61.89    844.42 

1769.6    68.73    69.56    895.56 

46 

48 

1674.1    61.48    62.14    846.13 

1772.9    68.99    69.82    897.27 

48 

50 

1677.4    61.72    62.39    847.83 

1776  2    69.25    70.09    898.98 

50 

52 

1680.7    61.96    6-^.64    849.53 

1779.5    69.50    70.36    900.69 

52 

54 

1684.0    62.21     62.89    851  24 

1782.8    69.76    70.62    902.40 

54 

56 

1687.3    62.45    63.14    852.94 

1786.1     70.02    70.89    904.10 

56 

58 

1690.6    62.70    63  39    854.65 

1789.4    70.28    71.17     905.81 

58 

60 

1693.9    62.94    63  64    856  35 

1792.7     70.54     71.42     907.52 

60 

FUNCTIONS  OF  A  ONE-DEGREE   CURVE       215 


18° 

19° 

/ 

L.  C.   M.    E.     T. 

L.  C.   M.    E.     T. 

/ 

0 

1792.7  70.54  71.42   907.52 

1891.5  78.58  79.65   958.86 

0 

2 

1796.0  70.80  71.69   909.23 

1894.8  78.86  79.94   960.57 

2 

4 

1799.3  71.06  71.96   910.94 

1898.1  79.13  80.22   962.30 

4 

6 

1802.6  71.33  72.23   912.65 

1901.3  79.41  80.51   964.00 

6 

8 

1805.9  71.59  72.50   914.36 

1904.6  79.68  80.79   965.72 

8 

10 

1809.2  71.85  72.77   916.07 

1907.9  79.96  81.08   967.43 

10 

12 

1812.5  72.12  73.04   917.78 

1911.2  80.24  81.37   969.15 

12 

14 

1815.7  72.38  73.31   919.49 

1914.5  80.51  81.65   970.86 

14 

16 

1819.0  72.64  73.58   921.20 

1917.8  80.79  81.94   972.58 

16 

18 

1822.3  72  91  73.85   922.91 

1921.0  81.07  82.22   974.29 

18 

20 

1825.6  73.17  74.12   924.63 

1924.3  81.35  82.51   976.01 

20 

22 

1828.9  73.43  74.39   926.34 

1927.6  81.63  82.80   977.72 

22 

24 

1832.2  73.70  74.67   928.05 

1930.9  81.91  83.09   979.44 

24 

26 

1835.5  73.97  74.94   929.76 

1934.2  82.20  83.38   981.15 

26 

28 

1838.8  74.24  75.22   931.47 

1937.5  82.48  83.67   982.86 

28 

30 

1842.1  74.51  75.49   933.18 

1940.7  82.76  83.97   984.58 

30 

32 

1845.4  74.77  75.77   934.89 

1944.0  83.05  84.26   986.30 

32 

34 

1848.7  75.04  76.04   936.60 

1947.3  83.33  84.55   988.02 

34 

36 

1852.0  75.31  76.32   938.32 

1950.6  83.61  84.84   989.74 

36 

38 

1855.3  75.58  76.59   940.03 

1953.9  83.90  85.13   991.46 

38 

40 

1858.6  75.85  76.87  941.74 

1957.2  84.18  85.43   993.18 

40 

42 

1861.9  76.12  77.14   943.45 

1960.4  84.47  85.73   994.90 

42 

44 

1865.1  76.39  77.42   945.16 

1963.7  84.75  86.02   996.62 

44 

46 

1868.4  76.67  77.70   946.88 

1967.0  85.04  86.32   998.34 

46 

48 

1871.7  76.94  77.98   948.59 

1970.3  85.32  86.61  1000.0 

48 

50 

1875.0  77.21  78.26   950.30 

1973.6  85.61  86.91  1001.8 

50 

52 

1878.3  77.49  78.53   952.01 

1976.9  85.90  87.21  1003.5 

52 

54 

1881.6  77.76  78.81   953.72 

1980.1  86.19  87.50  1005.2 

54 

5(5 

1884.9  78.03  79.09   955.44 

1983.4  86.47  87.80  1006.9 

56 

58 

1888.2  78.31  79.37   957.15 

19S6.7  86.76  88.09  1008.6 

58 

60 

1891.5  78.58  79.65   958.86 

1990.0  87.05  88.39  1010.4 

60 

20° 

21° 

/ 

L.  C.   M.    E.     T. 

L.  C.    M.    E.     T. 

/ 

0 

1990.0  87.05  88.39   1010.4 

2088.5   95.95   97.58  1062.0 

0 

2 

1993.3  87.34  88.69   1012.1 

2091.8   96.26   97.90  1063.7 

2 

4 

1996.6  87.63  88.99   1013.8 

2095.0   96.56   98.21  1065.4 

4 

6 

1999.8  87.92  89.29   1015.5 

2098.3   96.87   98.53  1067.2 

6 

8 

2003.1  88.21  89.59   1017.2 

2101.6   97.17   98.84  1068.9 

8 

10 

2006.4  88.50  89.89   1019.0 

2104.9   97.48   99.16  1070.6 

10 

18 

2009.7  88.79  90.19   1020.7 

2108.1   97.79   99.48  1072.4 

12 

14 

2013.0  89.08  90.49   1022.4 

2111.4   98.09   99.79  1074.1 

14 

16 

20163  89.37  90.79   1024.1 

2114.7  98.40  100.1   1075.8 

16 

18 

2019.5  89.66  91.09   1025.8 

2118.0   98.70  100.4   1077.5 

18 

20 

2022.8  89.96  91.40   1027.6 

2121.2   99.00  JOO.  7   1079.3 

20 

22 

2026.1  90.25  91.71   1029.3 

2124.5   99.30  J01.1   1081.0 

22 

24 

2029-4  90.55  92.01   1031.0 

2127.8   99.60  101.4   1082.7 

24 

26 

2032.7  90.85  92.32   1032.7 

2131.0   99.90  101.7   1084.4 

26 

28 

2036.0  91.15  92.62   1034.4 

2134.3  100.2   102.0   1086.2 

28 

30 

2039.2  91.45  92.93   1036.1 

2137.6  100.5   102.3   1087.9 

30 

32 

2042.5  91.74  93.24   1087.9 

2140.9  100.8   102.7   1089.6 

32 

34 

2045.8  92.04  93.54   1039.6 

2144.1  101.1   103.0   1091.3 

34 

36 

2049.1  92.34  93.85   1041.3 

2147.4  101.4   103.3   1093.1 

36 

38 

2052.4  92.64  94.15   1043.0 

2150.7  101.7   103.6   1094.8 

38 

40 

2055.7  92.94  94.46   1044.8 

2154.0  102.1   104.0   1096.5 

40 

42 

2058.9  98.24  94.78   1046.5 

2157.2  102.4   104.3   1098.3 

42 

44 

2062.2  93.54  95.09   1048.2 

2160.5  102.7   104.6   1100.0 

44 

46 

2065.5  93.84  95.40   1049.9 

2163.8  103.0   104.9   1101.7 

46 

48 

2068.8  94.14  85.71   1051.7 

2167.1  103.3   105.3   1103.4 

48 

50 

2072.1  94.44  96.03   1053.4 

2170.3  103.6   105.6   1105.2 

50 

52 

2075.4  94.74  96.34   1055.1 

2173.6  103.9   105.9   1106.9 

52 

54 

2078.6  95.04  96.65   1056.8 

2176.9  104.2   106.3   1108.6 

54 

56 

2081.9  95.34  96.96   1058.6 

2180.1  104.5   106.6   1110.3 

56 

58 

2085.2  95.64  97.27   1060.3 

2183.4  104.8   106.9   1112.1 

58 

60 

2088.5  95.95  97.58   1062.0 

2186.7  105.2  107.2   1118.8 

60 

216       FUNCTIONS  OF  A  ONE-DEGREE   CURVE 


22° 

23° 

L.  C.        M.         E.         T. 

L.  G.        M.          E.         T. 

0 

2186.7    105.2    107.2    1113.8 

2284.8    115.0    117.4    1165.8 

0 

2 

2190.0    105.6    107.6    1115.5 

2288.1     115.3    117.7    1167.5 

2 

4 

2193.2    105.9    107.9    1117.3 

2291.3    115.7    118.1    1169.2 

4 

6 

2196.5     106.2    108.2    1119.0 

2294.6    116.0    118.4    1171.0 

6 

8 

2199.8    106.5    108.6    1120.7 

2297.8    116.4    118.8    1172:7 

8 

10 

2203.0    106.8    108.9    1122.4 

2301.1     116.7    119.1    1174.4 

10 

12 

2206.3    107.1    109.2    1124.2 

2304.4    117.0    119.5    1176.2 

12 

14 

2209.6    107.4    109.6    1125.9 

2307.6    117.4    119.8    1177.9 

14 

16 

22129    107.7    109.9    1127.6 

2310.9    117.7     120.2    1179.7 

16 

18 

2216.1     108.0    110.2    1129.4 

2314.1     118.1     120.5-  1181.4 

18 

20 

2219.4    108.4    110.6    1131.1 

2317.4    118.4    120.9    1183.1 

20 

22 

2222.7    108.7    110.9    1132.8 

2320.7    118.7    121.2    1184.9 

22 

24 

2225.9    109.0    111.2    1134.6 

2323.9    119.1     121  6    1186.6 

24 

26 

2229.2    109.4    111.6     1136.3 

2327.2    119.4     121.9    1188.4 

26 

28 

2232.5    109.7    111.9    1138.0 

2330.4    119.8    122.3    1190.1 

28 

30 

2235.7    110.0    112.3    1139.7 

2333.7    120.1     122.6    1191.8 

30 

32 

2239.0     110.4     112.6     1141.5 

2337.0    120.4     123.0    1193.6 

82 

34 

22423    110.7    112.9     1143.2 

2340.2    120.8    123.3    1195.3 

34 

36 

2245.6    111.0    113.3    1144.9 

2343.5    121.1     123.7    1197.1 

36 

38 

2248.8    111.4    113.6    1146.7 

2346.7    121.5    124.1    1198.8 

38 

40 

2252.1     111.7    113.9    11J8.4 

2350.0    121.8    124.4    1200.5 

40 

42 

2255.4    112.0    114.3    1150.1 

2353.3     122.1     124.8    1202.3 

42 

44 

2258.6    112.3    114.6    1151.9 

2356.5    122.5    125.1     1204.0 

44 

46 

2261.9    112.7    115.0    1153.6 

2359.8    122.8     125.5    1205.8 

46 

48 

2265.2    113.0    115.3    1155.4 

2303.0    123.2     125.8    1207.5 

48 

50 

2268.4    113.3    115.7    1157.1 

2366.3    123.5     126.2    1209.2 

50 

52 

2271.7    113.7    116.0    1158.8 

2369.6    123.8    126.6    1211.0 

52 

54 

2275.0    114.0    116.3    1160.6 

2372.8    124.2    126.9    1212.7 

54 

56 

2278.3    114.3    116.7    116-2.3 

2376.1     124.5     127.3    1214.5 

56 

58 

2281.5    114.7    117.0    1164.0 

2379.3    124.9    127.6    1216.2 

58 

60 

2284.8    115.0    117.4    11G5.8 

2382.6     125.2    128.0    1218.0 

60 

24° 

25° 

L.  0.        M.          E.         T. 

L  C.        M.          E.         T. 

0 

23S2.6    125.2    128.0    1218.0 

2480.4    135.8     139.1     1270.3 

0 

2 

2385.9    125.5    128.4    1219.7 

2483.6    JJ6.2    139.5    1272.0 

2 

4 

2389.1     125.9    128.7    1221.4 

2486.9    136.5    139.9    1273.8 

4 

6 

2392.4     126.2     129.1     1223.2 

2490.1     136  9     140.3     1275.5 

6 

8 

2395.6    126.6    129.5    1224.9 

2493.4    137.2    140.6    1277.3 

8 

10 

2398.9    126.9    129.8    1226.7 

2496.6    137.6    141.0    1279.0 

10 

12 

2402.2    127.3     130.2    1228.4 

2499.9    138.0    141.4    1280.8 

12 

14 

2405.4    127.6    130.6    1230.2 

2503.1     1383     141  8    1282.5 

14 

1« 

2408.7    128.0    130.9    1231.9 

2506.4    138.7    142.2    1284.3 

16 

18 

2411.9    128.3    131.3    1233.6 

2509.6    139.0    142.5    1286.1 

18 

20 

2415.2    128.7    131.7    1235.4 

2512.9    139.4    142.9    1287.8 

20 

22 

2418.5    129.0    132.0    1237.1 

2516.1     139.8    143.3    1289.6 

22 

24 

2421.7    129.4    132.4    1233.9 

2519.4    140.1     143.7    1291.3 

24 

26 

2425.0    129.7    132.8    1240.6 

2522.6    140.5    144.1     1293.1 

26 

28 

2428.2    130.1     133  1     1242.4 

2525.9    140.8    144.5    1294.8 

28 

30 

2431.5    130.4    133.5    1244.1 

2529  1     141.2    144.9    1296.6 

30 

32 

2434.8    130.8    133.9    1215.8 

2532.4    141.6    145.3    1298.3 

32 

34 

2433.0    131.1     134.2    1247.6 

2535.6    142.0    145.6    1300.1 

34 

36 

2441.3    131.5     134.6    1249.3 

2538.9    142.3    146.0    1301.8 

36 

38 

2444.5    131.8    135.0    1251.1 

2542.1     142.7    146.4    1303.6 

38 

40 

2447.8    132.2    135.4    1252.8 

2545.4    143.1     146.8    1305.3 

40 

42 

2451.1     132.6    135.7    1254.6 

2548.6    143.5    147.2    1307.1 

42 

44 

2454.3    132.9    136.1     1256.3 

2551.9    143.8    147.6    1308.8 

44 

46 

2457.6    133.3    136.5    1258.1 

2555.1     144.2    148.0    1310.6 

46 

48 

2460.S    133.6    136.9    1259.8 

2558.4    144.5    148  4    1312.4 

48 

50 

2464.1     134.0    137.2    1261.5 

2561.6    144.9    148.8    1314.1 

50 

52 

2467.4    134.4    137.6    1263.3 

2564.9    145.3    149.2    1315.9 

52 

54 

2470.6    134.7    138.0    1265.0 

2568.1     145.7    149.5    1317.6 

54 

56 

2473.9    135.1     138.4    1266.8 

2571.4    146.0    149.9    1319.4 

56 

58 

2477.1     135.4    138.7    1268.5 

2574.6    146.4     150.3    1321.1 

58 

60 

2480.4    135.8    139.1     1270.3 

2577.9    146.8    150.7    1322.9 

60 

FUNCTIONS  OF  A  ONE-DEGREE  CURVE       217 


26° 

27* 

/ 

L.  C.       M.     .   E.          T. 

L.  C.       M.        E.          T. 

/ 

0 

2577.9    146.8    150.7     1322.9 

2675.3    158.3    162.8    1375.6 

0 

2 

2581.1     147.1    151.1    1324.6 

2678.5    158.6    163.2    1377.4 

2 

4 

2584.4    147.5    151.5    1326.4 

2681.8    159.0    163.7    1379.2 

4 

6 

2587.6    147.9    151.9    1328.1 

2685.0    159.4    164.1     1380.9 

6 

8 

2590.9    148.3    152.3    1329.9 

2688.2    159.8    164.5    1882  7 

8 

10 

2594.1     148.7    152.7    1331.6 

2691.5     160.2    164.9    1384.5 

10 

12 

2597.4     149.1     153.1     1333.4 

2694.7    160.6    165.3    1386.2 

12 

14 

2600.6    149.4    153.5    1335.2 

2698.0    181.0    165.7    1388.0 

14 

16 

2603.9    149.8    153.9    1336.9 

2701.2    161.4    1661     1389.8 

16 

18 

2607.1     150.2    154.3    1338.7 

2704.4    161.8    166.5    1391.5 

18 

20 

2610.4    150.6    154.7    1340.4 

2707.7    162.2    167.0    1393.3 

20 

22 

2613.6     151.0    155.1     1342.2 

2710.9    162.6    167.4    1395.0 

22 

24 

2616.9    151.4     155.5    1343.9 

2714.1     163.0    167.8    1396.8 

24 

26 

2620.1     151.7    155.9    1345.7 

2717.4    163.4    168.2    1398.6 

26 

28 

2623.4     152.1     156.3    1347.4 

2720.6    163.8    168.6    1400.3 

28 

30 

2626.6     152.5    156.7    1349.2 

2723.8     164.2    169.1     1402.1 

30 

32 

2629.8    152.9    157.1     1351.0 

2727.1     164.6    169.5    1403.9 

32 

34 

2633.1     153.3    157.5    1352.7 

2730.3    165.0    169.9    1405.6 

34 

36 

26363    153.7    157.9    1354.5 

2733.6    165.4    170.3    1407.4 

36 

38 

2639.6    154.0    158.3    1356.2 

2736.8    165.8    170.8    1409.2 

38 

40 

2642.8    154.4    158.7    1358.0 

2740.0    166.2    171.2    1410.9 

40 

42 

2646.1     154.8    159.1     1359.8 

2743.3    166.6    171.6    1412.7 

42 

44 

2649.3    155.2    159.5    1361.5 

2746.5    167.0    172.0    1414.5 

44 

46 

2652.6    155.6    160.0    1363.3 

2749.7    1674    172.5    1416.3 

46 

48 

2655.8    156.0    160.4    1365.1 

2753.0    167.8    172.9    1418.0 

48 

50 

2659.1     156.3    160.8    1366.8 

2756.2    168.2    173.3    1419.8 

50 

52 

2662.3    156.7    161.2    1368.6 

2759.5    168.6    173.7    1421.6 

52 

54 

2665.6    157.1     161.6    13~0.4 

2762.7    169.0    174.1     1423.3 

54 

56 

2668.8    157  5    162.0    13?2.1 

2765.9    169.4    174.6    1425.1 

56 

58 

2672.1     157.9    162.4    1373.9 

2769.2    169.8    175.0    1426.9 

58 

60 

2675.3    158.3    162.8    1375.6 

2772.4     170.2    175.4    1428.6 

60 

28° 

29° 

L.  C.        M.         E.          T. 

L.  C.        M.         E.         T. 

0 

2772.4    170.2    175.4    1428.6 

2869.4    182.5    188.5    1481.9 

0 

2 

2775.6    170.6    175.8    1430.4 

2872.6    182.9    189.0    1483.7 

2 

4 

2778.9    171.0    176.3     1432.2 

2875.8    183  3    189.4    1485.4 

4 

6 

2782.1     171.4     176.7    1434.0 

2879.1     183.7    189.9    1487.2 

6 

8 

2785.3    171.8    177.1     1435.7 

2882.3    184.2    190.3    1489.0 

8 

10 

2788.6    172.2    177.6    1437.5 

2885.5    184.6    190.8    1490.8 

10 

12 

2791.8    178.6    178.0    1439.3 

2888.7    185.0    191.2    1492.6 

12 

14 

2795.0    173.0    178.4    1441.1 

2892.0    185.4    191.7    1494.3 

14 

16 

2798.3    173.4    178.9    1442.8 

2895.2    185.8    192.1     1496.1 

16 

18 

2801.5    173.8    179.3    1444.6 

2898.4    186.3    192.5    1497.9 

18 

20 

2804.7    174.3    179.7    1446.4 

2901.6    186.7    193.0    1499.7 

20 

22 

2808.0    174.7    180.2    1448.2 

2904.8    187.1    193.5    1501.5 

22 

24 

2811.2    175.1     180.6    1449.9 

2908.1     187.5    193.9    1503.2 

24 

26 

2814.4    175.5    181.0    1451.7 

2911.3    188.0    194.4    1505.0 

26 

28 

2817.7     175.9    181  5    1453.5 

2914.5    188.4    194.8    1506.8 

28 

30 

2820.9    176.3    181.9    1455.2 

2917.7    188.8    195.3    1S08.6 

30 

32 

2824.1     176.7    182.3    1457.0 

2921.0    189.2    195.7    1510.4 

32 

34 

2827.4    177.1     182.8    1458.8 

2924.2    189.7    196.2    1512.1 

34 

36 

2830.6    177.5    183.2    1460.6 

2927.4    190.1    196.7    1513.9 

36 

38 

2833.8    177.9    183.6    1462.3 

2930.6    190.5    197.1     1515.7 

38 

40 

2837.1     178.4    184.1    1464.1 

2933.9    190.9    197.6    1517.5 

40 

4-2 

2840.3    178.8    184.5    1465.9 

2937.1    191.4    198.0    1519.3 

42 

44 

2843  5     179.2    185.0    1467.7 

2940.3    191.9    198.5    1521.0 

44 

46 

2846.8    179.6    185.4    1469.5 

2943.5    192.4     198.9    1522.8 

46 

48 

2850.0    1800    185.9    1471.2 

2946.8    192.8    199.4    1524.6 

48 

50 

2853.2    180.4    186.3    1473.0 

2950  0    193.2    199.8    1526.4 

50 

52 

2856.5    180.8    186.8    1474.8 

2953.2    193.6    200.3    1528.2 

52 

54 

2859.7    181.2    187.2    1476.6 

2956.4    194.0    200.8    1530.0 

54 

56 

2862.9    181.6    187.6    1478.3 

2959.6    194.4    201.2    1531.7 

56 

58 

2866.2    1820    188.1     14801 

2962.9    194.8    201.7    1533.5 

58 

60 

2869.4     1825    188.5     1481.9 

2966.1     195.2    202.1     15353 

60 

218       FUNCTIONS  OF  A  ONE-DEGREE  CURVE 


80° 

31° 

/ 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

0 

2966.1  195.2  202.1  1535.3 

3062.6  208.4  2163  1589.0 

0 

2 

2969.3  195.6  202.6  1537.1 

3065.8  208.8  216.8  1590.8 

2 

4 

2972.5  190.1  203.1  1538.9 

3069.0  209.3  217.2  1592.6 

4 

6 

2975.7  196.5  -203.5  1540.7 

3072.2  209.7  217.7  1594.4 

6 

8 

2^79  0  197.0  X'04.0  1542.5 

3075.4  210.2  218.2  1596.2 

8 

10 

2982.2  197.4  204.5  1544  3 

3078.6  210.6  218.7  1598.0 

10 

12 

2985.4  197.8  204.9  154G.O 

3081.8  211.1  219.2  1599.8 

12 

14 

2988.6  198.2  205.4  1547.8 

3085.0  211.5  219.6  1601.6 

14 

16 

2991.8  198.6  2059  1549.6 

3088.3  212.0  220.1  1603.4 

16 

18 

2995.0  199.1  2063  1551.4 

3091.5  212.4  220.6  1605.2 

18 

20 

2998.3  199.5  206.8  1553.2 

3094.7  212.9  221.1  1607.0 

20 

22 

3001.5  199.9  207.3  1555.0 

3097.9  213.3  221  6  1608.8 

22 

24 

3004.7  200.4  207.7  1556.8 

3101.1  213.8  222.1  1610.6 

24 

26 

3007.9  200.8  208.2  1558.6 

3104.3  214.2  222.6  1612.4 

26 

28 

3011.1  201.3  208.7  1560.4 

3107.5  214.7  223.0  1614.2 

28 

30 

3014.3  201.7  209.1  1562.2 

3110.7  215.1  223.5  1616.0 

30 

32 

3017.6  202.1  209.6  1564.0 

3113.9  215.6  224.0  1617.8 

32 

34 

3020.8  202.6  210.1  1565.7 

3117.1  216.0  224.5  1619.6 

34 

36 

3024.0  203.0  210.5  1567.5 

3120.3  216.5  225.0  1621.4 

36 

38 

3027.2  203.5  211.0  1569.3 

3123.5  216.9  225.5  1623.2 

38 

40 

3030.4  203.9  211  5  1571.1 

3126.7  217.4  226.0  1625.0 

40 

42 

3033.6  204.3  212  0  1572.9 

3129.9  217.8  226.5  1626.8 

42 

44 

3036.9  204.8  212.4  1574.7 

3133.1  218.3  227.0  1628.6 

44 

46 

3040.1  205.2  212.9  1576.5 

3136.4  218.7  227.5  1630.5 

46 

48 

3043.3  205.7  213.4  1578.3 

3139.6  219.2  228.0  1632.3 

48 

50 

3046.5  206.1  213.9  1580.1 

3142.8  219.6  228.4  1634.1 

50 

52 

3049.7  206.5  214.4  1581.9 

3146.0  220.1  228.9  1635.9 

52 

54 

3052.9  207  0  214.8  1583.7 

3149.2  220.5  229.4  1637.7 

54 

56 

3056.2  207.4  215.3  1585.5 

3152.4  221.0  2-)9.9  1639.5 

56 

58 

3059.4  207.9  215.8  1587.2 

3155.6  221.5  230.4  1641.3 

58 

60 

3062.6  208.4  216  3  1589.0 

3158.8  2-J2.0  230.9  1643.1    60 

32° 

33° 

L.  C.   M.    E.    T. 

L.  C.    M.    E.    T. 

o 

3158.8  222.0  230.9  1G43.1 

3254.9  236.0  246.1  1697  3 

0 

2 

3162.0  2225  231.4  1644.9 

3258  1  236.4  246.6  1699.1 

2 

4 

3165.2  222.9  231.9  1646.7 

3261.3  236.9  247.1  17009 

4 

6 

3168.4  223.4  232.4  1648.5 

3264.5  237.4  247.7  1702.7 

6 

8 

3171.6  223.8  232.9  1650.3 

3267  7  237.9  248.2  1704.5 

8 

10 

3174.8  224.3  233.4  1652.1 

3270.8  238.4  248.7  1706.4 

10 

12 

3178.0  224  8  233.9  1653.9 

3274  0  238  9  249.2  1708.2 

12 

14 

3181.2  225.2  234.4  16557 

3277  2  239  3  249.7  1710.0 

14 

16 

3184.4  225.7  234.9  1657.5 

32804  239.8  250.2  1711.8 

16 

18 

3187.6  226.1  235.4  1659.3 

3-J83.6  240.3  250.8  1713.6 

18 

20 

3190.8  226.6  235.9  1661.1 

3286.8  240.8  251.3  1715.5 

20 

22 

3194.0  227.1  236.4  1662.9 

3290  0  241.2  251.8  1717.3 

22 

24 

3197  2  227.5  236  9  1664.7 

3293.2  241  7  252.3  1719.1 

24 

26 

3200.4  228.0  237.4  1666  5 

3296.4  242.2  252.9  1720.9 

26 

28 

3203.6  228.4  237.9  1668.3 

3299.6  242.7  253.4  1722.7 

28 

30 

3206.8  228  9  238.4  1670.1 

3302.7  2432  253.9  1724.6 

30 

32 

3210.0  229.4  239.0  1671.9 

3305  9  243.6  254.4  1726.4 

32 

34 

3213.2  229  8  239.5  1673.7 

3309  1  244.1  255.0  1728.2 

34 

36 

3216.5  230.3  240.0  1675.5 

3312.3  244.6  255.5  1730.0 

36 

88 

8219.7  230.7  240.5  1677.4 

33155  245.1  256.0  1731.8 

38 

40 

3222.9  231  2  241.0  1679.2 

3318.7  245.6  256  5  1733.6 

40 

42 

32261  231.7  241.5  1681.0 

3321.9  2460  2571  1735.5 

42 

44 

3629.3  232.2  242.0  1682.8 

3325.1  246.5  257.6  1737.3 

44 

46 

3232.5  232.6  242.5  1684.6 

3328.3  247.0  258.1  1739.1 

46 

48 

3235.7  233.1  243.0  16864 

3331.5  247  5  258  6  1740.9 

48 

50 

3238.9  233.5  243.5  1688.2 

3334.6  248.0  259.2  1742.7 

50 

52 

3242.1  234.0  244  1  1690.0 

3337  8  248.4  259.7  1744.6 

52 

54 

3245.3  234.5  244.6  1691.8 

3341.0  248.9  260  2  1746.4 

54 

56 

3248.5  2350  245.1  1693.7 

3344-2  249.4  260,8  1748.2 

56 

58 

3251.7  235.5  245.6  1695  5 

3347.4  249.9  261.3  1750.0 

58 

60 

3254.9  236.0  246.1  1697.3 

3350-6  250.4  261  .B  1751.8 

60 

FUNCTIONS  OF  A  ONE-DEGREE  CURVE       219 


34° 

35° 

/ 

L.  O.        M.         E.         T. 

L.  C.        M.         E.         T. 

0 

3350.6    250.4    261.8    1751.8 

3446.1    265.2    278.1     1806.7 

0 

2 

3353.8    250.8    262.3    1753.7 

3449.3    265.7    278.6    1808.5 

2 

4 

3357.0    251.2    262.9    1755.5 

3452.5    266.2    279.2    1810.3 

4 

6 

3360.1    251.7    263.4    1757.3 

3455.6    266.7    279.7    1812.2 

6 

8 

3363.3    252.2    204.0    1759.1 

3458.8    267.2    280.3    1814.0 

8 

10 

3366.5    252.7    264.5    1761.0 

3462.0    267*7    280.8    1815.8 

10 

12 

3369.7    253.2    265.0    1762.8 

3465.2    268.2    281.4    1817.7 

12 

14 

3372.9    253.7    265.6    1764.6 

3468.3    268.7    281.9    1819.5 

14 

16 

3376.1     254.2    266.1      766.4 

3471.5    269.2    282.5    1821.3 

16 

18 

3379.2    254.7    266.7      768.3 

3474.7    269.7    283.0    1823.2 

18 

20 

3382.4    255.2    267.2    1770.1 

34V7.9    270.2    283.6    1825.0 

20 

22 

3385.6    255.7    267.7      771.9 

3481.0    270.7    284.2    1826  8 

22 

24 

3388.8    256.2    268.3      773.7 

3484.2    271.2    284.7    18'28.7 

24 

26 

339-2.0    256.7    268.8      775.6 

3487.4    271.7    285.3    1830.5 

26 

28 

3395.2    257.2    269.3      777.4 

3490.6    272.2    285.9    1832.3 

28 

30 

3398.3    257.7    269.9      779.2 

3493.7    272.7    286.4    1834.2 

30 

32 

3401.5    258.2    270.4      781.0 

3496.9    273.2    287.0    1836.0 

32 

34 

3404.7    258.7    271.0      782.9 

3500.1    273.  Y    287.5    1837.8 

34 

36 

3407.9    259  2    271.5      784.7 

3503.3    274.2    288.1     1839  7 

36 

38 

3411.1    259.7    2720    1786.5 

3506.5    274.7    288.7    1841.5 

38 

40 

3414.3    260.2    272.6    1788.4 

3509.6    275.2    289,2    1843.4 

40 

42 

3417.4    260.7    273.1     1790.2 

3512.8    275.7    289.8    1845.2 

42 

44 

3420.6    261.2    273.7    1792.0 

3516.0    2762    290.4     1847.1 

44 

46 

3423.8    261.7    274.2    1793.9 

3519.2    276.7    290.9    1848.9 

46 

48 

3427.0    262.2    274.8    1795.7 

35-22.3    277.2    291.5    1850.7 

48 

50 

3430.2    262.7    275.3    1797.5 

3525.5    277.7    292.0     185'2.6 

50 

52 

3433.4    263.2    275.9    1799.3 

3528.7    278.2    292.6    1854.4 

52 

54 

3436.5    263.7    276.4    1801.2 

3531.9    278.7    293.2    1856.3 

54 

56 

3439.7    264.2    277.0    1803.0 

3535.0    279.2    293.7    1858.1 

56 

58 

3442.9    264.7    277.5    1804.8 

3538.2    279.8    294.3    1859.9 

58 

60 

3446.1     265.2    278.1     1806.7 

3541.4    280.4    294.9    1861.8 

60 

36° 

37° 

/ 

L.  C.        M.         E.          T. 

L.  C.        M.         E.          T. 

/ 

0 

3541.4    280.4    294.9    1861.8 

3636.3    296.1    312.3    1917.3 

0 

2 

3544.6    280  9    295.4    1863.6 

3639.5    296.6    312.8    1919.1 

2 

4 

3547.7    281.4    296.0    1865.5 

3642  6    297  1     313.4    1921.0 

4 

6 

3550.9    281.9    296.6    1867.3 

3645.8    297.7    314.0    1922.8 

6 

8 

3554.0    282.5    297.2    1869.2 

3648.9    298.2    314.6    1924.7 

8 

10 

3557.2    283.0    297.7    1871.0 

3652.1    298.7    315.2    1926.5 

10 

12 

3560.4    283.5    298.3    1872.9 

3655.2    299.3    315.8    1928.4 

12 

14 

3563.5    284.0    298.9    1874.7 

3658.4    299.8    316.4    1930  2 

14 

16 

3566.7    284.6    299.5    1876.5 

3661.6    300.3    317.0    1932.1 

16 

18 

3569.9    285.1    300.0    187'8.4 

3061.7    300.9    317.5    1933.9 

18 

20 

3573.0    285.6    300.6    1880.2 

3667.9    301.4    318.1     1935.8 

20 

22 

3576.2    286.1     301.2    1882.1 

3671.0    301.9    318.7    1937.6 

22 

24 

3579.4    286.7    301.8    1883.9 

3674.2    302.5    319.3    1939.5 

24 

26 

3582.5    2S7.2    302.3    1885.8 

3677.3    303.0    319.9    1941.3 

26 

28 

3585.7    287.7    302.9    1887.6 

3680.5    303.5    320.5    1943.2 

28 

30 

3588.8    288.2    303.5    1889.5 

3683.6    304.1     321.1     1945.0 

30 

32 

3592.0    288.8    304.1     1891.3 

3686.8    304.6    321.7     1946.9 

32 

34 

3595.2    289.3    304.6    1893.2 

3690.0    305.1     322.3     1948.8 

34 

36 

3598.3    289  8    305.2    1895.0 

3693.1     305.7    322.9    1950.6 

36 

38 

3601.5    290.3    305.8    1896.9 

3696.3    306.2    323.5    1952.5 

38 

40 

3604.7    290.9    306.4    1898.7 

3699.4    306.7    324.2    1954.4 

40 

42 

3607.8    291.4    307.0    1900.6 

3702.6    307.3    324.8    1956.2 

42 

44 

3611.0    291.9    307.5    1902.4 

3705.7    307.8    325.4    1958.1 

44 

46 

3614.1    292.4    308.1     1904.3 

3708.9    308.3    326.0    1960.0 

46 

48 

3617.3    293.0    308.7    1906.1 

3712.1     308.9    3266    1961.8 

48 

50 

3620.5    293.5    309.3    1908.0 

3715  2    309.4    327  2    1963.  V 

50 

52 

3623.6    294.0    309.9    1909.8 

3718.4    309.9    327.8    1965.5 

52 

54 

3626.8    294.5    310.5    1911.7 

3721  5    310.5    3-28.4     1967.4 

54 

56 

3630.0    295.1    311.1     1913.5 

3724.7    311.0    329.0    1969.3 

56 

58 

3633.1     295.6    311.7    1915.4 

3727.8    311.6    329.6    1971.1 

58 

60 

3636.3    296.1     312.3    1917.3 

3731.0    312.2    330.2    1973.0 

60 

220       FUNCTIONS  OF  A  ONE-DEGREE  CURVE 


38° 

39° 

1 

/ 

L.  C.   M.   E.    T. 

L.  C.   M    E.    T. 

0 

3731.0  312.2  330.2  1973.0 

3825.5  328-7  348.7  2029.1 

0 

2 

3734.1  312.7  330.8  1974.9 

3828.6  329.2  349  3  2031.0 

2 

4 

3737.3  313.3  331.4  1976-7 

3831.8  329.8  349.9  2032  9 

4 

6 

3740.4  313.8  332.0  1978.6 

3834.9  330-3  350.6  2034.7 

6 

8 

3743.6  314.4  332.6  1980.5 

3838.0  330.9  351.2  2036.6 

8 

10 

3746.7  314.9  333.2  1982.3 

3841.2  331  5  351.8  2038.5 

10 

12 

3749.9  315.5  333.8  1984.2 

3844  3  332.0  352-4  8040.4 

12 

14 

37530  3160  334-5  1986.1 

3847.4  332.6  358.1  2042  3 

14 

16 

37562  316.6  335.1  1987.9 

3850.6  333.2  353.7  2044.1 

16 

18 

3759.3  317.1  335  7  1989.8 

3853  7  333  7  354.3  2046.0 

18 

20 

3762  5  317.7  336  3  1991.7 

3856.8  334.3  354.9  2047.9 

20 

22 

3765.6  318  2  336  9  1993-6 

3860  0  334.9  355.6  2049.8 

22 

24 

3768.8  318.8  337.5  1995.4 

3863.1  335.4  356.2  2051.7 

24 

26 

3771  9  319.3  338.1  1997.3 

3866.2  336-0  356.9  2053.5 

26 

28 

3775.1  319.9  338.7  1999.2 

3869.4  336.6  357  5  2055.4 

28 

30 

3778.2  3204  339.4  2001.0 

3872.5  337.1  358.1  2057-3 

30 

32 

3781.4  321.0  340.0  2002.9 

3875  6  337  7  358.8  2059.2 

32 

34 

3784  5  321.5  340-6  2004.8 

3878.8  338.3  359  4  2061.1 

34 

36 

3787.7  322.1  341.2  2006.6 

3881.9  338.8  360.1  2063.0 

36 

38 

37908  322.6  341.8  2008.5 

3885.0  339  4  360.7  2064-8 

38 

40 

3794.0  338.2  342.4  2010  4 

3888.2  340.0  361.3  2066.7 

40 

4-2 

37971  323.7  343.1  2012.3 

3891.3  340.5  362.0  2068  .6 

42 

44 

3800.3  324.3  343.7  2014.1 

3894  4  341  1  362.6  2070.5 

44 

46 

3803.4  324.8  344.3  2016  0 

3897.6  341.7  363.3  2072.4 

46 

48 

3806.6  325.4  344.9  2017.9 

3900  7  342  2  363.9  2074  2 

48 

50 

3809.7  325  9  345.6  2019.7 

3903.8  342.8  364.5  2076  1 

50 

52 

3812.9  3265  346.2  2021.6 

3907.0  343  4  365.2  2078.0 

52 

54 

3816.0  3270  346.8  2023.5 

3910.1  343.9  365.8  2079.9 

54 

56 

3819.2  327.6  347.4  2025  4 

3913.2  344.5  3665  2081.8 

56 

68 

3822.3  323.1  3481  2027.2 

3916.4  345  1  367.1  2083.7 

58 

60 

3825.5  328.7  348-7  2029.1 

3919  5  345.6  367.7  2085-5 

60 

40° 

41° 

L.  C.   M.    E.     T. 

L.  C.   M.    E.    T. 

0 

3919.5  345.6  367.7  2085.5 

4013.4  362.9  387.4  2142.3 

0 

2 

3922.6  346.1  368.4  2087  4 

4016.5  G63.4  888.1  2144.2 

2 

4 

3925.8  346  7  309  0  2089.3 

4019.6  364.0  388.8  2146.1 

4 

6 

3928.9  347.2  369.7  2091  2 

4022.7  364.5  389.4  2148.0 

6 

8 

3932  0  347.8  370.3  20H3.1 

4025.9  365.1  390  1  2149.9 

8 

10 

3935.1  348.4  371.0  2095  0 

4029.0  365.6  390.7  2151.9 

10 

12 

3938.3  348.9  371  6  2096  9 

4032.1  366.2  391.4  2158.8 

12 

14 

3941.4  349.5  372.3  2098.8 

4035.2  366.8  392.1  2)55.7 

14 

16 

3944  5  350.1  372.9  2100.7 

4038.3  367.4  392.7  2157.6 

16 

18 

8947.7  350.7  373.6  2102  6 

4041.4  368.0  393.4  2159.5 

18 

20 

3950.8  351.3  374.3  2104.5 

4044.6  368.6  394.1  2161.4 

20 

2:2 

3953.9  351  8  374.9  2106.3 

4047.7  369.2  394.7  2163.3 

22 

24 

3957.1  352.4  375.6  2108.2 

4050.8  369.8  395.4  2165.2 

24 

26 

3960.2  353.0  376.2  2110.1 

4053.9  370.4  396.1  2167.1 

26 

28 

3963.3  353.6  376.9  21120 

4057.0  371.0  396.8  2169.0 

28 

30 

3966.4  354.2  377.5  2113  9 

4060.1  371.6  397.5  2170.9 

30 

32 

3969.6  354.7  378.2  2115  8 

4063.3  372.2  398.1  2172.8 

32 

34 

3972.7  355.3  378.8  2117  7 

4066.4  372.8  398.8  2174.7 

34 

36 

3975.8  355.9  379.5  2119  6 

4069.5  373.4  399.5  2176.6 

36 

38 

3979.0  356.5  380  1  2121.5 

4072.6  374.0  400.2  2178.5 

38 

40 

3982.1  357.1  380.8  2123.4 

4075.7  374.6  400.9  2180.4 

40 

4-> 

3985.2  357  6  381.4  2125  3 

4078.8  375.2  401.5  2182.4 

42 

44 

3988.4  358.2  382.1  2127.2 

4082.0  375.8  402.2  2184.8 

44 

46 

3991  5  358  8  382  8  2129.1 

4085.1  376.4  402.9  2186.2 

46 

48 

3994.6  359.4  383.4  2131.0 

4088.2  377.0  403.6  2188.1 

48 

50 

3997  7  360  0  384.1  2132.9 

4091.3  377.6  404.3  2190.0 

50 

52 

4000  9  360  5  384.8  2134.7 

4094.4  378.2  404.9  2191.9 

52 

54 

4004.0  361  1  385.4  2136.0 

4097.5  378.8  405.6  2198.8 

54 

56 

4007  1  361.7  386.1  2138.5 

4100.7  379.4  406.3  2195.7 

56 

58 

40103  362.3  3868  21404 

4103.8  380.0  407.0  2197.6 

58 

60 

4013.4  362.9  387.4  2142.3 

4106.9  380.6  407.7  2199.5 

60 

FUNCTIONS  OF  A  ONE-DEGREE  CURVE       221 


42° 

43° 

& 

L.  C.   M.   E.     T. 

L.  C.   M.   E.     T. 

0 

4106.9  380.6  407.7  2199.5 

4200.1  398.7  428.6  2257.1 

0 

2 

4110.0  381.2  408.3  2:201.4 

4203.2  399.3  429.3  2259.0 

2 

4 

4113.1  381.8  409.0  2-J03.3 

4206.3  399.9  430.0  2261.0 

4 

6 

4116.2  382.4  409.7  2205.3 

4209.4  400.5  430.7  2262.9 

6 

8 

4119.3  383.0  410.4  2207.2 

4212.5  401.1  431.4  2264.8 

8 

10 

4122.4  383.6  411.1  2209.1 

4215.6  401.7  432.1  2266.7 

10 

12 

4125.5  384.2  411.8  2211.0 

4218.7  402.4  432.8  2268.7 

12 

14 

4128.6  384.8  412.5  2212.9 

4221.8  403.0  433.5  2270.6 

14 

16 

4131.8  38r  4  413.2  2214.9 

4224.9  403.6  434.2  2272.5 

16 

18 

4134.9  38b  0  413.9  2216.8 

4228.0  404.2  434.9  2274.5 

18 

20 

4138.0  386  6  414.6  2218.7 

4231.1  404.8  435.6  2276.4 

20 

22 

4141.1  387.2  415.3  2220.6 

4234.2  405.4  436.3  S278.3 

22 

24 

4144.2  387.8  416.0  2222.5 

4237.3  406.1  437.0  2280.2 

24 

26 

4147.3  388.4  416.6  2224.4 

4240.4  406.7  437.8  2282.2 

26 

28 

4150.4  389.0  417.3  2226.4 

4243.5  407.3  438.5  2284.1 

28 

30 

4153.5  389.6  418.0  2228.3 

4246.5  407.9  439.2  2286.0 

30 

32 

4156.6  390,2  418.7  2230.2 

4249.6  408.5  439.9  2288.0 

32 

34 

4159.7  390.8  419.4  223'?.  1 

4252.7  409.1  440.6  2289.9 

34 

36 

4162.8  391.4  420.1  2234.0 

4255.8  409.8  441.4  2291.8 

36 

38 

4165.9  392.0  420.8  2236.0 

4258.9  410.4  442.1  2293.8 

38 

40 

4169.0  392.6  421.5  2237.9 

4262.0  411.0  442.8  2295.7 

40 

42 

4172.1  393.2  422.2  2239.8 

4265.1  411.6  443.5  2297.7 

42 

44 

4175.2  393.8  422.9  2241.7 

4268.2  412.2  444.2  2299.6 

44 

46 

4178.4  394.4  423.6  2243.6 

4271.3  412.8  445.0  2301.5 

46 

48 

4181.5  395.0  424.3  2245.6 

4274.4  413.5  445.7  2303.5 

48 

50 

4184.6  395.6  425.0  2247.5 

4277.5  414.1  446.4  2305.4 

50 

52 

4187.7  396.2  425.7  2249.4 

4280.6  414.7  447.1  2307.3 

52 

54 

4190.8  396.8  426.4  2251.3 

4283.7  415.3  447.8  2309.3 

54 

56 

4193.9  397.4  427.1  2253.3 

4286.8  415.9  448.6  2311.2 

56 

58 

4197.0  398.0  427.8  2255.2 

4289.9  416.5  449.3  2313.1 

58 

60 

4-,'00.1  398.7  428.6  2257.1 

4293.0  417.2  450.0  2315.1 

60 

440 

45° 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

f 

0 

4293.0  417.2  450.0  2315.1 

438").  5  436.2  472.1  2373.4 

0 

2 

4296  1  417.8  450.7  2317.0 

4388.6  436.8  472.9  2375.4 

2 

4 

4-299.2  418.4  451.5  2319.0 

4391.7  437.5  473.6  2377.3 

4 

6 

4302.2  419.1  452.2  2320.9 

4394.7  438.1  474.4  2379.3 

6  - 

8 

4305.3  419.7  452  9  2322.8 

4397.8  438.8  475.1  2381.2 

8 

10 

4308.4  420.3  453.7  2324.8 

4400.9  439.4  475.9  2383.2 

10 

VI 

4311.5  421.0  454.4  2326.7 

4404.0  440.0  476.6  2385.2 

12 

14 

4314.6  421.6  4f,5.1  2328.7 

4407.0  440.7  477.4  2387.1 

14 

16 

4317.7  422.2  455.9  2330.6 

4410.1  441.3  478.1  2389.1 

16 

18 

4320.7  422.9  456.6  2332.6 

4413.2  442.0  478.9  2891.0 

18 

20 

4323.8  423  5  457.3  2334.5 

4416.3  442.6  479.6  2393.0 

20 

22 

4326.9  424.1  458.1  2336.4 

4419.3  443.2  480.4  2394.9 

22 

24 

4330.0  424.8  458.8  2338.4 

4422.4  443.9  481.1  2396.9 

24 

26 

4333.1  425  4  459.5  2340  3 

4425.5  444.5  481.9  2398.8 

26 

28 

4336  2  426.0  460.3  2342.3 

4428.6  445.2  4S2.6  2400.8 

28 

30 

4339.2  426.7  461  0  2344.2 

4431.6  445.8  483.4  2402.8 

30 

3-4 

4342.3  427.3  461.7  2346.1 

4434.7  446.4  484.2  2404.7 

32 

34 

4345.4  427  9  462.5  2348.1 

4437.8  447.1  484.9  2406.7 

34 

36 

4348.5  4286  463.2  2350.0 

4440.9  447.7  485.7  2408.6 

36 

38 

4351.6  429.2  463.9  2352.0 

4444.0  448.3  486.5  24106 

sL 

40 

4354.7  429.8  464.7  2353.9 

4447.0  448.9  487.2  2412.6 

40 

42 

4357.7  430.5  485.4  2355.9 

4450.1  449,5  488.0  2414.5 

42 

44 

4360.8  431.1  466.2  2357.8 

4453  2  450.2  488.7  2416.5 

44 

46 

4363.9  431.7  466.9  2359.8 

4456.3  450.8  489.5  2418.5 

46 

48 

4367.0  432.4  467.7  2361.7 

4459.3  451.5  490.3  2420.4 

48 

50 

4370.1  433.0  468  4  2363.7 

4462.4  452.1  491.0  2422.4 

50 

52 

4373.2  433.6  469.1  2365.6 

4465.5  452.7  491  8  2424.4 

52 

54 

43762  4343  469.9  2367.6 

4468.6  453  4  492.5  2426.3 

54 

56 

43^9.3  434.9  470.6  2369.5 

4471.6  454.1  493.3  2428.3 

56 

58 

4382.4  435  b  471.4  28*1.5 

4474.7  454.8  494.1  2430.2 

58 

60 

4385.5  436.2  472.1  2373.4 

4477.8  455.5  494.8  2432.2 

60 

222      FUNCTIONS  OF  A  ONE-DEGREE  CURVE 


46° 

47° 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

0 

4477.8  455.5  494.8  2432.2 

4569.7  475.2  518.3  2491.5 

0 

2 

4480.9  456.1  495.6  2434.2 

4572.7  475.9  519.0  2493.4 

2 

4 

4483.9  456.8  496.5  2436.1 

4575.8  476.5  519.8  2495.4 

4 

6 

4487.0  457.4  497.2  2438.1 

4578.8  477.2  520.6  2497.4 

6 

8 

4490.0  458.1  497.9  2440.1 

4581.9  477.8  521.4  2499.4 

8 

10 

4493.1  458.7  498.7  2442.1 

4584.9  478.5  522.2  2501.4 

10 

12 

4496.2  459.4  499.5  2444.0 

4588.0  479.2  523.0  2503.4 

12 

14 

4499.2  460.0  500.3  2446.0 

4591.0  479.8  523.8  2505.4 

14 

16 

4502.3  460.7  501.0  2448.0 

4594.1  480.5  524.6  2507.3 

16 

18 

4505.4  461.3  501.8  2449.9 

4597.1  481.1  525.4  2509.3 

18 

20 

4508.4  462.0  502.6  2451.9 

4600.2  481.7  526.2  2511.3 

20 

22 

4511.5  462.7  503.4  2453.9 

4603.2  482.3  527.0  2513.3 

22 

24 

4514.6  463.3  504.1  24559 

4606.3  483.0  527.8  2515.3 

24 

26 

4517.6  464.0  504.9  2457.8 

4609.3  483.7  528.6  2517.3 

26 

28 

4520.7  464.6  505.7  2459.8 

4612.4  484.3  529.4  2519.3 

28 

30 

4523.7  465.3  506.5  2461.8 

4615.4  485.0  530.2  2521.2 

30 

32 

4526.8  466.0  507.3  2463.  S 

4618.5  485.7  531.0  2523.2 

32 

34 

4529.9  466.6  508.0  2465.7 

4621.5  486.3  531.8  2525.2 

34 

36 

4532.9  467.3  508.8  2467.7 

4624.6  487.0  532.6  2527.2 

36 

88 

4536.0  467.9  509.6  2469.7 

4627.6  487.7  533.4  2529.2 

38 

40 

4539.1  468.6  510.4  2471.7 

4630.7  488.4  534.2  2531.2 

40 

42 

4542.1  469.3  511.1  2473.6 

4633.7  489.1  535.0  2533.2 

42 

44 

4545.2  469.9  511.9  2475.6 

4636.8  489.8  535.8  2535.2 

44 

46 

4548.2  470.6  512.7  2477.6 

4639.8  490.5  536.6  2537.2 

46 

48 

4551.3  471.2  513.5  2479.6 

4642.9  491.2  537.4  2539.2 

48 

50 

4554.4  471.9  514.3  2481.6 

4645.9  491.9  53S.2  2541.2 

50 

52 

4557.4  472.6  515.1  2483.5 

4649.0  492.6  539.0  2543.1 

52 

54 

4560.5  473.2  515.9  2485.5 

4652.0  493.3  539.8  2545.1 

54 

56 

4563.6  473.9  516.7  2487.5 

4655.1  494.0  540.6  2547.1 

56 

58 

4566.6  474.5  517.5  2489.5 

4658.1  494.7  541.4  2549.1 

58 

60 

4569.7  475.2  518.3  2491.5 

4661.2  495.4  542.3  2551.1 

60 

48° 

4U° 

/ 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

0 

4661.2  495.4  542.3  2551. 

4752.3  515.9  567.0  2611.3 

0 

2 

4664.2  496.0  543.1  2553. 

4755.3  516.5  567.8  2613.3 

2 

4 

4667.3  496.7  543.9  2555. 

4758.4  517.2  568.7  2615.3 

4 

6 

4670.3  497.4  544.7  2557. 

4761.4  517.9  569.5  2617.3 

6 

8 

4673.3  498.1  545.5  2559. 

4764.4  518.6  570.3  2619.3 

8 

10 

4676.4  498.8  546.4  2561. 

4767.4  519.3  571.2  2621.4 

10 

12 

4679.4  499.4  547.2  2563. 

4770.5  520.0  572.0  2623.4 

12 

14 

4682.5  500.1  548.0  2565. 

4773.5  520.7  572.8  2625.4 

14 

16 

4685.5  500.8  548.8  2567. 

4776.5  521.4  573.7  2627.4 

16 

18 

4688.5  501.5  549.6  2569. 

4779.6  522.1  574.5  2629.4 

18 

20 

4691.6  502.2  550.5  2571. 

4782.6  522.8  575.3  2631.4 

20 

22 

4694.6  502.8  551.3  2573. 

4785.6  523.5  576.2  2633.5 

22 

24 

4697.6  503.5  552.1  2575. 

4788.7  524.2  577.0  2635.5 

24 

26 

4700.7  504.2  552.9  2577.1 

4791.7  524.9  577.9  2637.5 

26 

28 

4703.7  504.9  553.7  2579. 

4794.7  525.6  578.7  2639.5 

28 

30 

4706.7  505.6  554.6  2581. 

4797.7  5','6.3  579.6  2641.5 

30 

32 

4709.8  506.2  555.4  2583.1 

4800.8  527.0  580.4  2643.5 

32 

34 

4712.8  506.9  556.2  2585.1 

4803.8  527.7  581.3  2645.6 

34 

36 

4715.9  507.6  557.0  2587.2 

4806.8  528.4  582.1  2647.6 

36 

38 

4718.9  508.3  557.8  2589.2 

4809.9  529.1  583.0  2649.6 

38 

40 

4721.9  509.0  558.7  2591.2 

4812.9  529.8  583.8  2651.6 

40 

42 

4725.0  509.6  559.5  2593.2 

4815.9  530.5  584.7  2653.7 

42 

44 

4728.0  510.3  560.3  2595.2 

4819.0  531.2  585.5  2655.7 

44 

46 

4731.0  511.0  561.2  2597.2 

4822.0  531.9  586.4  2657.7 

46 

48 

4734.1  511.7  562.0  2599.2 

4825.0  532.6  587.2  2659.7 

48 

50 

4737.1  512.4  562.8  2601.2 

4828.0  533.3  588.1  2661.8 

50 

52 

4740.2  513.1  563.7  2603.2 

4831.1  534.0  5S8.9  2663.8 

52 

54 

4743.2  513.8  564.5  2605.2 

4834.1  534.7  589.8  2665.8 

54 

56 

4746.2  514.5  565.3  2607.2 

4837.1  535.4  590.6  2667.8 

56 

58 

4749.3  515.2  566.2  2609.3 

4840.2  536.1  591.5  2669  9 

58 

60 

4752.3  515.9  567.0  2611.3 

4843.2  536.8  592.4  2671.9 

60 

FUNCTIONS  OF  A  ONE-DEGREE  CURVE       223 


60° 

61° 

L.  C.   M.   E.    T. 

L.  C.   M.   E.    T. 

0 

4843.2  536.8  592.4  2671.9 

4933  6  558.2  618.5  2733.0 

0 

2 

4846.2  537.5  593.2  2673  9 

4936.6  558.9  619.3  2735.1 

2 

4 

4849.2  538.2  594.1  2676.0 

4939.6  559.7  620.2  2737.1 

4 

6 

4852.2  538.9  594.9  2678.0 

4942.6  560.4  621.1  2739.2 

6 

8 

4855.2  539.6  595.8  2680.0 

4945.6  561.1  622.0  2741.2 

8 

10 

4858.3  540.3  596.7  2682.1 

4948.6  561.8  622.9  2743.3 

10 

re 

4861.3  541.0  597.5  2684.1 

4951.6  562.5  623.7  2745.3 

12 

14 

4864.3  541.7  598.4  2686.1 

4954.6  563.3  624.6  2747.4 

14 

16 

4867.3  542.4  599.3  2688.2 

4957.6  584.0  625.5  2749.4 

16 

18 

4870.3  543.1  600.1  2690.2 

4960.6  564.7  626.4  2751.5 

18 

80 

4873.3  543.9  601.0  2692.3 

4963.6  565.4  627.3  2753.5 

20 

22 

4876.3  544.6  601.9  2694.3 

4966.6  566.2  628.2  2755.6 

22 

24 

4879.4  545.3  602.7  2696.3 

4969.6  566.9  629.9  2757.7 

24 

26 

4882.4  546.0  603.6  2698.4 

4972.6  567.6  630.0  2759.7 

26 

28 

4885.4  546.7  604.5  2700.4 

4975.6  568.3  630.9  2761.8 

28 

30 

4888.4  547.4  605.3  2702.4 

4978.6  569.1  631.8  2763.8 

30 

32 

4891.4  548.1  606.2  2704.5 

4981.6  569.8  632.7  2765.9 

32 

34 

4894.4  548.8  607.0  2706.5 

4984.6  570.5  633.6  2767.9 

34 

33 

4897.4  549.5  607.9  2708.6 

4987.   571.2  G34.5  2770.0 

36 

38 

4900.4  550.2  608.8  2710.6 

4990.   572.0  635.3  2772.0 

38 

40 

4903.5  551.0  609.7  2712.6 

4993.   572.7  636.2  2774.1 

40 

42 

4906.5  551.7  610.5  2714.7 

4996.7  573.4  637.1  2776.2 

42 

44 

4909.5  552.4  611.4  2716.7 

4999.   574.1  638.0  2778.2 

44 

46 

4912.5  553.1  612.3  2718.8 

5002.   574.9  638.9  2780.3 

46 

48 

4915.5  553.8  613.2  2720.8 

5005.   575.6  639.8  2782.3 

48 

50 

4918.5  554.5  614.1  2722.8 

5008.   576.3  640.7  2784.4 

50 

52 

4921.5  555.2  614.9  2724.9 

5011.7  577.0  641.6  2786.4 

52 

54 

4924.6  555.9  615.8  2726.9 

5014.7  577.8  642.5  2788.5 

54 

56 

4927.6  556.6  616.7  2729.0 

5017.7  578.5  643.4  2790.6 

56 

58 

4930.6  557.4  617.6  2731.0 

5020.7  579.2  644.3  2792.6 

58 

60 

4933.6  558.2  618.5  2733.0 

5023.7  579.9  645.2  2794.7 

60 

52° 

53° 

L.  C.   M.   E.     T. 

L.  C.    M.   E.     T. 

0 

5023.7  579.9  645.2  2794.7 

5113.5  602.0  672.7  2856.9 

0 

5026.7  580.6  646.1  2796.8 

5116.5  602.8  673.7  2858.9 

2 

4 

5029.7  581.3  647.0  2798.8 

5119.4  603.5  674.6  2861.0 

4 

6 

5032.7  582.1  647.9  2800.9 

5122.4  604.3  675.5  2863.1 

6 

8 

5035.7  582.8  648.9  2803.0 

5125.4  605.0  676.4  2865.2 

8 

10 

5038.7  583.5  649.8  2805.0 

5128.4  605.8  677.4  2867.3 

10 

12 

5041.7  584.3  650.7  2807.1 

5131.3  606.5  678.3  2869.4 

12 

14 

5044.7  585.0  651.6  2809.2 

5134.3  607.3  6T9.2  2871.5 

14 

16 

5047.7  585.7  652.5  2811.2 

5137.3  608.0  680.2  2873.5 

16 

18 

5050.7  586.5  653.4  2813.3 

5140.3  608.8  681.1  2875.6 

18 

20 

5053.6  587.2  654.3  2815.4 

5143.2  609.5  682.0  2877.7 

20 

22 

5056.6  587.9  655.2  2817.4 

5146.2  610.3  683.0  2879.8 

22 

24 

5059.6  588.7  656.2  2819.5 

5149.2  611.0  683.9  2881.9 

24 

26 

5062.6  589.4  657.1  28-'1.6 

5152.1  611.8  684.9  2884.0 

26 

28 

5065.6  590.1  658.0  2823.6 

5155.1  612.5  685.8  2886.1 

28 

30 

5068.6  590.9  658.9  2825.7 

5158.1  613.3  686.7  2888.1 

30 

32 

5071.6  591.6  659.8  2827.8 

5161.1  614.0  687.7  2890.2 

32 

34 

5074.6  592.3  660.7  2829.8 

5164.0  614.8  688.6  2892.3 

34 

36 

5077.6  593.1  661.6  2831.9 

5167.0  615.5  689.6  2894.4 

36 

38 

5080.6  593.8  662.5  2834.0 

5170.0  616.3  690.5  2896.5 

38 

40 

5083.6  594.5  663.5  2836.1 

5173.0  617.0  691.5  2898.6 

40 

42 

5086.6  595.3  664.4  2838.2 

5175.9  617.8  692.4  2900.7 

42 

44 

5089.6  596.0  665.3  2840.2 

5178.9  61S.5  693.4  2902.8 

44 

46 

5092.6  596.7  666.2  2842.3 

5181.9  619.3  694.3  2904.9 

46 

48 

5095.6  597.5  667.2  2844.4 

5184.9  620.1  695.3  2907.0 

48 

50 

5098.6  598.2  668.1  2846.5 

51b7.8  620.8  696.2  2909.1 

50 

52 

5101.6  598.9  669.0  '2848.  5 

5190.8  621.5  697.1  2911.2 

52 

54 

5104.6  599.7  669.9  2850.6 

5193.8  622.3  698.1  2913.3 

54 

56 

5107.6  600.4  670.9  2852.7 

5196.7  623.0  699.0  2915.4 

56 

58 

5110.6  601.2  671.8  2854.8 

5199.7  6',>3.8  700.0  2917.5 

58 

60 

5113.5  602.0  672.7  2856.9 

S20-J.7  624.6  700.9  2919.5 

60 

224       FUNCTIONS  OF  A  ONE-DEGREE  CURVE 


54° 

65° 

/ 

L.C.   M.    E.    T. 

L.  C.   M.    E.    T. 

0 

5202.7  624.6  700.9  2919.5 

5291.7  647.4  7'29.9  2982.8 

0 

2 

5205.7  625.4  701.9  2921.6 

5294.6  648.1  730.9  2984.9 

2 

4 

5208.6  626.1  702.8  2923.8 

5297.6  648.9  731.9  2987.1 

4 

6 

5211.6  626.9  703.8  2925.9 

5300.5  649.6  732.9  2989.2 

6 

8 

5214.6  627.6  704.8  2928.0 

5303.5  650.4  733.8  2991.3 

8 

10 

5217.5  628.4  705.7  2930.1 

5306.4  651.2  734.8  2993.4 

10 

12 

5220.5  629.2  706.7  2932.2 

5309.4  652.0  735.8  2995  5 

12 

14 

5223.5  629.9  707.7  2934.3 

5312.3  652.7  736.8  2997.7 

14 

16 

5226.4  630.7  708-6  2936.4 

5315.3  653.5  737.8  2999.8 

16 

18 

5229.4  631.4  109.6  2938.5 

5318.2  654.3  733.7  3001.9 

18 

20 

5232.4  632.2  710.5  2940.6 

53-21.2  655.1  739.7  3004.0 

20 

22 

5235.3  633.0  711.5  2942.7 

5324.1  655.8  740.7  3006.2 

22 

24 

5238.3  633.7  712.5  2944.8 

5327.1  656.6  741.7  3008.3 

24 

26 

5241.3  634.5  713.4  2946.9 

53.30.0  057.  4  742.7  3010.4 

26 

28 

5244.2  635.2  714.4  2949.0 

5333.0  658.2  7'43.7  3012.5 

28 

30 

5247.2  636.0  715.3  2951.1 

5335.9  658.9  744.7  3014.7 

30 

32 

5250.2  636.8  716.3  2953.2 

5338.8  659.7  745  7  3016.8 

32 

34 

5253.1  637.5  717.3  2955.3 

5341.8  660.5  746.7  3018.9 

34 

36 

5256.1  638.3  718.2  2957.5 

5344.7  661.3  747.7  3021.1 

36 

38 

5259.1  639.0  719.2  2959.6 

5347.7  662.0  748.7  3023.2 

38 

40 

5262.0  639.8  720.2  2961.7 

5350.6  66-2.8  749.7  3025.3 

40 

42 

5265.0  640.6  721.1  2963.8 

5353.6  6636  750.7  3027.5 

42 

44 

5268.0  641.3  722.1  2965.9 

5356.5  664.4  751.7  3029.6 

44 

46 

5270.9  642.1  723.1  2968.0 

5359.5  665.1  752.6  3031.7 

46 

48 

5-273.9  642.8  724.1  2970.1 

5362.4  665.9  753.6  3033.8 

48 

50 

5276.9  643.6  725.0  2972.2 

5365.4  666.7  754.6  3036.0 

50 

52 

5279.8  644.4  726.0  2974.4 

5368.3  667.5  755.6  3038.1 

52 

54 

5282.8  645.1  727.0  2976.5 

5371.3  668.3  756.6  3040.2 

54 

56 

5285.8  645.9  728.0  2978.6 

5374.2  669.1  757.6  3042.4 

56 

58 

5288.7  646.6  729.0  2980.7 

5377.2  669.9  758.6  3044.5 

58 

60 

5291.7  647.4  729.9  2382.8 

5380.1  670.7  759.6  3046.6 

60 

66° 

57° 

' 

L.  C.   M.    E.    T. 

L.C.   M.    E.    T. 

0 

5380.1  670.7  759.6  3046.6 

5468.2  694.4  790.2  3111.1 

0 

2 

5383.0  671.4  760.6  3048.8 

5471.1  695.2  791.2  3113.3 

2 

4 

5386.0  672.2  761.6  3050.9 

5474.0  696.0  792.2  3115.4 

4 

6 

5388.9  672.9  762.7  3053.1 

5477.0  696.8  793.3  3117.6 

6 

8 

5391.8  673.7  763.7  3055.2 

5479.9  697.6  794.3  3119.7 

8 

10 

5394.8  674.4  764.7  3057.4 

5482.8  698.4  795.3  3121.9 

10 

12 

5:597.7  675.2  705.7  3059.5 

5485.7  699.2  796.3  3124.1 

12 

14 

5400.7  676.0  7G6.7  3061.6 

5488.  7  700.0  797.4  3126.2 

14 

16 

5403.6  676.8  767.7  3063.8 

5491.6  700.8  798.4  3128.4 

16 

18 

5406.5  677.6  768.7  3065.9 

5494.5  701.6  799.4  3130.6 

18 

20 

5409.5  678.4  769.7  3068.1 

5497.4  702.4  800.5  3132.7 

20 

22 

5412.4  679.2  770.8  3070.2 

5500.3  703.2  801.5  3134.9 

2'2 

24 

5415.3  680.0  771.8  3072.4 

5503.3  704.0  802.6  3137.0 

24 

26 

5418.3  680.8  772.8  307'4.5 

5506.2  704.8  803.6  3139.2 

26 

28 

5421.2  681.6  773.8  3076.6 

5509.1  705.6  804.7  3141.4 

28 

30 

5424.1  682.4  774.8  3078.8 

5512.0  706.4  805.7  3143.5 

30 

3-2 

5427.1  683.2  775.8  3080.9 

5515.0  707.2  806.8  3145.7 

32 

34 

5430.0  684.0  776.8  3083.1 

5517.9  708.0  807.8  3147.9 

34 

36 

5433.0  684.8  777.8  3085.2 

5520.8  708.8  808.8  3150.0 

36 

38 

5435.9  685.6  778.9  3087.4 

55:23.7  709.6  809.9  3152.2 

38 

40 

5438.8  686.4  779.9  3089.6 

55-26.7  710.4  810.9  3154.4 

40 

42 

5441.8  687.2  780.9  3091.7 

5529.6  711.2  812.0  3156.6 

42 

44 

5444.7  688.0  781.9  3093.9 

5532.5  712.0  813.0  3158.7 

44 

46 

5447.6  688.8  783.0  3096.0 

5535.4  712.8  814.1  3160.9 

46 

48 

5450.6  689.6  784.0  3098.2 

5538.4  713.6  815.1  3163.1 

48 

50 

5453.5  690.4  785.0  3100.3 

5541.3  714.4  816.2  3165.3 

50 

52 

5456.5  691.2  786.0  3102.5 

5544.2  715.2  817.2  3167.4 

52 

54 

5459.4  692.0  787.1  3104.6 

5547.1  716.0  818.3  3169.6 

54 

56 

5462.3  69-2.8  788.1  3106.8 

5550.0  716.8  819.3  3171.8 

56 

58 

5465.3  693.6  789.1  3108.9 

5553.0  717.6  820.4  3174.0 

58 

60 

5468.2  694.4  790.2  3111.1 

5555.9  718.4  821.4  3170.  1 

60 

FUNCTIONS  OF  A  ONE-DEGREE   CURVE       225 


1 

68° 

69° 

1 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

0 

5555.9  718.4  821.4  3176.1 

5643.1  742.8  853.5  3241.9 

0 

2 

5558.8  719.2  822.5  3178.3 

5646.0  743.6  854.6  3244.1 

2 

4 

5561.7  720.0  823.5  3180.5 

5648.9  744  4  855.7  3246.3 

4 

c 

5564.6  720.8  824.6  3182.7 

5651.8  745.3  856.8  3248.5 

6 

8 

5567.5  721.6  825.7  3184.9 

5654.7  746.1  857.9  3250.7 

8 

10 

5570.4  722.4  826.7  3187.1 

5657.6  746.9  859.0  8252.9 

10 

12 

5573.3  723.2  827.8  3189.2 

5660.5  747.7  860.0  3255.1 

12 

14 

5576.2  724.0  828.9  3191.4 

•  5663.4  748.6  861.1  3257.3 

14 

16 

5579.2  724.8  829.9  3193.6 

5666.3  749.4  862.2  3259.5 

16  ' 

18 

5582.1  725.6  831.0  3195.8 

5669.2  750.2  863.3  3261.7 

18 

20 

5585.0  726.5  832.1  3198.0 

•  5672.1  751.1  864.4  3263.9 

20 

22 

5587.9  727.3  833.1  3200.2 

5675.0  751.9  865-5  3266.1 

22 

24 

5590.8  728.1  834.2  3202.4 

5677.9  752.7  866.6  3268  3 

24 

26 

5593.7  728.9  835.3  3204.5 

5680.8  753.5  867.7  3270.5 

26 

28 

5596.6  729.7  836.3  3206.7 

5683.7  754.4  868.8  3272.7 

28 

30 

5599.5  730  5  837.4  3208.9 

5686  5  755.2  869.9  3274.9 

30 

32 

5602.4  731.3  838.4  3211.1 

5689.4  756.0  871.0  3277.1 

82 

34 

5605.3  732.1  839.5  3213.3 

5692.3  756.9  872.1  3279.4 

34 

36 

5608.2  732.9  840.6  3215.5 

5695.2  757.7  873.2  3281  6 

36 

38 

5611.1  733.7  841.6  3217.7 

5698  1  758.5  874.3  3283.8 

38 

40 

5614.0  734.6  842.7  3219.9 

5701.0  7594  875.4  3286.0 

40 

42 

5616.9  735.4  843.8  3222.1 

5703.9  760.2  876.5  3288.2 

42 

44 

5619.8  730.2  844.9  3224.3 

5706.8  761  0  877.6  3290.5 

44 

4G 

5622.8  737.0  846.0  3226.5 

5709  7  761.9  878.7  3292.7 

46 

4S 

5625.7  7378  847.0  3228  7 

5712.6  762.7  879.8  3294.9 

48 

50 

5628.6  738.6  848.1  32309 

5715.5  763.5  880.9  3297.1 

50 

58 

5631.5  739.4  849.2  3233.1 

5718.4  764.4  882.0  3299.3 

52 

54 

5634.4  740.2  850.3  3235.3 

5721.3  765.2  883.1  3301.5 

54 

50 

5637.3  741  0  851.4  3237.5 

5724.2  766.0  884.2  3303.8 

56 

58 

5040.3  741.9  85-.>.5  3239.7 

5727.1  766.8  885.3  3306.0 

58 

60 

5643.1  742.8  853.5  3241.9 

5730.0  767.7  886.4  3308.2 

60 

60° 

61° 

L.  C.   M,    E.    T. 

L.  C.   M.    E.    T. 

0 

5730.0  767.7  886.4  3308.2 

5816.4  792.9  920.2  3375.2 

0 

2 

5732.9  768.5  887.5  3310.4 

5819.3  793.7  921.4  3377.4 

2 

4 

5735.8  769.4  888.7  3312.7 

5822.1  794.6  922.5  3379.7  . 

4 

6 

5738.6  770.2  889.8  3314.9 

5825.0  795.4  923  6  3381.9 

6 

8 

5741.5  771.1  890.9  3317.1 

5827.9  796.3  924.8  3384.2 

8 

10 

5744  4  771.9  892.0  3319.3 

5830.7  797.1  935.0  3386.4 

10 

12 

5747.3  772.7  893.1  3321.6 

5833.6  798.0  927.1  3388.7 

12 

14 

5750.2  773.6  894.3  3323.8 

5836.5  798.8  928.2  3390.9 

14 

16 

5753.0  774.4  895.4  3326.0 

5839.3  799.7  929.3  3393.2 

16 

18 

5755.9  775.3  896.5  3328.3 

5842.2  800.5  930.5  3395.4 

18 

20 

5758.8  776.1  897.6  3330.5 

5845.1  801.4  931.6  33S7.7 

20 

22 

5761.7  776.9  898.8  3332.7 

5847.9  802.2  932.8  8399.9 

22 

24 

5764.6  777.8  899.9  3334.9 

5850.8  803.1  933.9  3402.2 

24 

26 

5767.4  778.6  901.0  3337.2 

5853.7  803.9  935.1  3404.4 

26 

28 

5770.3  779.5  902.1  3339.4 

5856.5  804.8  936.3  3406.7 

28 

30 

5773.2  780.3  903.2  3341.6 

5859.4  805.6  937..  4  3408.9 

30 

32 

5776.1  781.1  904.4  3343.9 

5862.3  806.5  938.6  3411.2 

32 

34 

5779.0  782.0  905.5  3346.1 

5865.1  807.3  939.7  3413.5 

34 

36 

5781.8  782.8  906.6  3348.3 

5868.0  808.2  940.9  3415.7 

36 

38 

5784.7  783.7  907.7  3350.6 

5870.9  809.0  942.1  3418.0 

38 

40 

5787.6  784.5  908.8  3352.8 

5873.7  809.9  943.2  3420.3 

40 

42 

5790.5  785.3  910.0  3355.0 

5876.6  810.7  944.4  3422.5 

42 

44 

5793.4  786.2  911.1  3357.3 

5879.5  811.6  945.5  3424.8 

44 

46 

5796.2  787.0  912.3  3359.5 

5882.3  812.4  946.7  3427.1 

46 

48 

5799.1  787.9  913.4  3361.8 

5885.2  813.3  947.8  3429.3 

48 

50 

5802.0  788.7  914.5  3364.0 

5888.1  814.1  949.0  3431.6 

50 

52 

5804.9  789.5  915.7  3366.2 

5890.9  815.0  950.2  3433.9 

52 

54 

5807.8  790.4  916.8  3368.5 

5893.8  815.8  951.3  3436.1 

54 

56 

5810.6  791.2  918.0  3370.7 

5896.7  816.7  952.5  3438.4 

56 

58 

5813.5  792.1  919.1  3373.0 

5899.5  817.5  953.6  3440.7 

58 

60 

5816.4  792.9  920.2  3375.2 

5902.4  818.4  954.8  3442.9 

60  j 

226       FUNCTIONS  OF  A  ONE-DEGREE   CURVE 


62° 

63» 

/ 

L.  C.   M.    E.    T. 

L.  C.    M.    E.    T. 

/ 

0 

5902.4  818.4  954.8  3442.9 

5987.8  844.4   990.3  3511.3 

0 

2 

5905.2  819.3  956.0  3445.2 

5990.6  845.3   991.5  3513.6 

2 

4 

5908.1  820.1  957.2  3447.5 

5993.5  846.2   992.7  3515.9 

4 

6 

5910.9  821.0  958  3  3449.7 

5096.3  847.1   993.9  3518.2 

6 

8 

5913.8  821.8  959.5  3452.0 

5999.1  847.9   995.1  3520.5 

8 

10 

5916.6  822.7  960.7  3454.3 

6002  0  848.8   996.3  3522.8 

10 

12 

5919.5  823.6  961.9  3456.6 

6004.8  849.7   997.5  3525.1 

12 

14 

5922.3  824.4  963.0  3458.8 

6007.7  850.6   998.7  3527.4 

14 

16 

5925.2  825.3  9G4.2  3461.1 

6010.5  851.4   999.9  3529.7 

16 

18 

5928.0  826.1  965.4  3463.4 

6013.3  852.3  1001.1  3532.0 

18 

20 

5930.9  827.0  966.6  3465.7 

6016.2  853.2  1002.3  3534.3 

20 

22 

5933.7  827.9  967.8  3107.9 

6019.0  854.1  1003.5  3536.6 

22 

24 

5936.6  828.7  968.9  3470.2 

6021.8  854.9  1004.7  3538.9 

24 

26 

5939.4  829.6  970.1  3472.5 

6024.7  855.8  1005.9  3541.2 

26 

28 

5942.3  830.4  971.3  3474.7 

6027.5  856.7  1007.1  3543.5 

28 

30 

5945.1  831.3  972.5  3477.0 

6030.3  857.6  1008.4  3545.8 

30 

82 

5947.9  832.2  973.6  3479.3 

6033.2  858.4  1009.6  3548.1 

32 

34 

5950.8  833.0  974.8  3481.6 

6036.0  859.3  1010.8  3550.4 

34 

36 

5953.6  833.9  976.0  3483.9 

6038.9  860.2  1012.0  3552.7 

36 

U8 

5956.5  834.7  977.2  3486.2 

6041.7  861.1  1013.2  3555.0 

38 

40 

5959.3  835.6  978.4  3488.5 

6044.5  861.9  1014.5  3557.3 

40 

42 

5962  2  836.5  979.6  3490.7 

6047.4  862-8  1015.7  3550.6 

42 

44 

5965.0  837.4  980.8  3493.0 

6050.2  863.7  1016.9  356-J.O 

44 

46 

5967.9  838.3  982.0  3495.3 

6053.0  864.6  1018.1  3564.3 

46 

48 

5970.7  839.1  983.2  3497.6 

C055.9  865.4  1019.3  3566.6 

48 

50 

5973.6  840.0  984.4  3499.9 

6058.7  866.3  1020.6  3568.9 

50 

52 

5976.4  840.9  985.5  3502.2 

6061.6  867.2  1021.8  3571.2 

52 

54 

5979.3  841.7  986.7  3504.5 

6064.4  868.1  1023.0  3573.5 

54 

56 

5982.1  842.6  987.9  3506.8 

6067.2  868.9  1024.2  3575.8 

56 

58 

5985.0  843.5  989.1  3509.0 

6070.1  869.8  1025.4  3578.1 

58 

60 

5987.8  844.4  990.3  3511.3 

6072.9  870.7  1026.7  3580.4 

60 

64°: 

66° 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

0 

6072.9  870.7  1026.7  3580.4 

6157.5  897.3  1064.0  3650.4 

0 

2 

6075.7  871.5  1027.9  3582.8 

6160.3  898.2  1065.2  3652.8 

2 

4 

•6078.5  872.4  1029.2  3585.1 

6163.1  899.1  1066.5  3655.1 

4 

6 

6081.4  873.3  1030.4  3587.4 

6165.9  900.0  1067.7  3657.5 

6 

8 

6084.2  874.2  1031.7  3589.7 

6168.7  900.9  1069.0  3659.8 

8 

10 

6087.0  875.1  1032.9  3592.1 

6171.5  901.8  1070.2  3662.2 

10 

12 

6089.8  875.9  1034.1  3594.4 

6174.3  902.7  1071.5  3664.5 

12 

14 

609:2.6  876.8  1035.4  3596.7 

6177.1  903.6  1072.7  3666.9 

14 

16 

6095.5  877.7  1036.6  3599.1 

6179.9  904.5  1074.0  3669.2 

16 

18 

0098.3  878.6  1037.9  3601.4 

6182.7  905.4  1075.2  3671.6 

18 

20 

6101.1  879.5  1039.1  3603.7 

6185.5  906.3  1076.6  3673.9 

20 

22 

6103.9  880.3  1040.3  3606.0 

6188.3  907.2  1077.8  3676.2 

22 

24 

6106.7  881.2  1041.  G  3608.4 

6191.1  908.1  1079.1  3678.6 

24 

26 

6109.6  882.1  1042.8  3610.7 

6193.9  909.0  1080.4  3680.9 

26 

28 

6112.4  883.0  1044.1  3613.0 

6196.7  909.9  1081.7  3683.3 

28 

30 

6115.2  883.9  1045.3  3615.3 

6199.5  910.8  1083.0  3685.6 

30 

32 

6118.0  884.7  1046.5  3617.7 

6202.3  911.7  1084.2  3688.0 

32 

34 

6120.8  885.6  1047.8  3620.0 

6205.1  912.6  1085.5  3690.4 

34 

36 

6123.7  886.5  1049.0  3622.3 

6208.0  913.5  1086.8  3692.7 

36 

38 

6126.5  887.4  1050.3  3624.7 

6210.8  914.4  1088.1  3695.1 

38 

40 

6129.3  888.3  1051.5  3627.0 

6213.6  915.3  1089.4  3697.4 

40 

42 

6132.1  889.2  1052.7  3629.4 

6216.4  916.2  1090.6  3699.8 

42 

44 

6134.9  890.1  1054.0  3631.7 

6219.2  917.1  1091.9  3702.2 

44 

46 

6137.8  891.0  1055.2  3634.0 

6222.0  918.0  1093.2  3704.5 

46 

48 

6140.6  891.9  1056.5  3636.4 

6224.8  918.9  1094.5  3706.9 

48 

50 

6143.4  892.8  1057.7  3638.7 

6227.6  919.8  1095.8  3709-3 

50 

52 

6146.2  893.7  1059.0  3641.1 

6230.4  920.7  1097.0  3711.6 

52 

54 

6149.0  894.6  1060.2  3643.4 

6233.2  921.6  1098.3  3714.0 

54 

56 

6151.9  895.5  1061.5  3645.7 

6236.0  922.5  1099.6  3716.3 

56 

58 

6154.7  896.4  10(52.7  3648.1 

6238.8  923.4  1100.9  3718.7 

58 

60 

6157.5  897.3  1064.0  3650.4 

6241.6  924.3  1102.2  3721.1 

60 

FUNCTIONS  OF  A  ONE-DEGREE  CURVE       227 


66° 

67« 

L.  C.   M.    E.     T. 

L.  C.   M.    E.    T. 

0 

6241.6  924.3  1102.2  3721.1 

6325.2  951.8  1141.5  3792.6 

0 

2 

6244.4  925.2  1103.5  3723.4 

6328.0  952.7  1142.8  3795.0 

2 

4 

6247.2  926.1  1104.8  3725.8 

6330.7  953.6  1144.1  3797.4 

4 

6 

«6250.0  927.0  1106.1  3728.2 

C333.5  954.5  1145.4  3799.8 

6 

8 

6252.7  927.9  1107.4  3730.6 

6336.3  955.5  1146.7  3802.2 

8 

10 

6255  5  928.8  1108.7  3732.9 

6339.0  956.4  1148.1  3804.6 

10 

12 

6258.3  929.8  1110.0  3735.3 

6341.8  957.3  1149.4  3807.0 

12 

14 

6201.1  930.7  1111.8  3737.7 

6344.6  958.2  1150.7  3809.4 

14 

16 

6263.9  931  6  1112.6  3740.1 

6347.4  959.2  1152.0  3811.8 

16 

18 

6266.7  (J32.5  1113.9  3742.4 

6350.1  960.1  1153.3  3814.2 

18 

20 

6269.5  933.4  1115.2  3744.8 

6352.9  961.0  1154.7  3816.6 

20 

22 

6272.3  934.3  1116.5  3747.2 

6355.7  961.9  1156.0  3819.0 

22 

24 

6275  0  935.3  1117.8  8749.6 

6358.4  962.9  1157.4  3821.4 

24 

26 

6277  8  936.2  1119.1  3751.9 

6361.2  963.8  1158.7  3823.8 

26 

28 

6280.6  937.1  1120.4  3754.3 

6364.0  964.7  1160.1  3826.2 

28 

30 

6283.4  938.0  1121.7  3756.7 

6366.7  965.6  1161.4  3828.6 

30 

32 

6286.2  938.9  1123.0  3759.1 

6369.5  966.6  1162.8  3831.0 

32 

34 

6289.0  939.8  1124.3  3761.5 

6372.3  907.5  1164.1  3833.4 

34 

36 

6291.8  940.8  1125.6  3763.9 

6375.1  968.4  1165.5  3835.9 

36 

38 

6294.5  941.7  1126.9  3766.3 

6377.8  969.3  1166.8  3838.3 

38 

40 

6297.3  942  6  1128.3  3768.7 

6380.6  970.3  1168.2  3840.7 

40 

42 

6300.1  943.5  1129.6  3771.0 

6383.4  971.2  1169.5  3843.1 

42 

44 

6302.9  944.4  1130.9  3773.4 

6386.1  972.1  1170.9  3845.5 

44 

46 

6305.7  945.3  1132.2  3775.8 

6388.9  973.0  1172.2  3847.9 

46 

48 

6308.5  946.3  1133.5  3778.8 

6391.7  974.0  1173.6  3850.4 

48 

50 

6311.3  947.2  1134.9  3780.6 

6394.4  974.9  1174.9  3852.8 

50 

52 

6314.1  948.1  1136.2  3783.0 

6397.2  975.8  1176.3  3855.2 

52 

51 

6316.8  949.0  1137.5  3785.4 

6400.0  976.8  1177.6  3857.6 

54 

56 

6319.6  949.9  1138.8  3787.8 

6402.8  977.7  1179.0  3860.0 

56 

58 

6322.4  950.8  1140.1  3790.2 

6405.5  978.6  1180.3  3862.5 

58 

60 

6325.2  951.8  1141.5  3792.6 

6408.3  979.6  1181.6  3864.9 

60 

68° 

69° 

L  C.   M.    E.    T. 

L.  C.    M.    E.    T. 

0 

6408.3  979.6  1181.6  3864.9 

6491.1  1007.7  1222.9  3938.1 

0 

2 

6411.1  980.5  1183.0  3867.3 

6493.8  1008.7  1224.3  3940.6 

2 

4 

6413.8  981.4  1184.4  3869.7 

6496.6  1009.6  1225.7  3943.0 

4 

6 

6416.6  982.4  1185.7  3872.2 

6499.3  1010.6  1227.1  3945.5 

6 

8 

6419.3  983.3  1187.1  3874.6 

6502.1  1011.5  1228.5  3947.9 

8 

10 

6422.1  984.2  1188.5  3877.0 

6504.8  1012.5  1229.9  3950.4 

10 

12 

6421.9  985.2  1189.8  3879.5 

6507.5  1013.4  1231.3  3952.9 

12 

14 

6427.6  986.1  1191.2  3881.9 

6510.3  1014.4  1232.7  3955.3 

14 

10 

6430.4  987.0  1192.6  3884.3 

6513.0  1015.3  1234.1  3957.8 

18 

18 

6433.1  988.0  1193.9  3886.8 

6515.8  1016.3  1235.5  3960.2 

18 

20 

6435.9  988.9  1195.3  3889.2 

6518.5  1017.2  1236.9  3962.7 

20 

22 

6438.7  989.8  1196.7  3891.6 

6521.2  1018.2  1238.3  3965.2 

22 

24 

6141.4  990.8  1198.0  3894.1 

6524.0  1019.1  1239.7  3967.6 

24 

26 

6444.2  991.7  1199.4  3896.5 

6526.7  1020.1  1241.1  3970.1 

26 

28 

64415.  9  992.6  1200.8  3898.9 

6529.5  1021.0  1242.5  3972.5 

28 

30 

6149.7  991.6  1202.1  3901.4 

6532.2  1022.0  1243.9  3975.0 

30 

32 

6452.5  994.5  1203.5  3903.8 

6534.9  1022.9  1245.3  3977.5 

32 

34 

6455.2  995.4  1204.9  3906.3 

6537.7  1023.9  1246.7  3980.0 

34 

36 

6458.0  996.4  1206.2  3908.7 

6640.4  1024.8  1248.1  3982.4 

36 

38 

6460.7  997.3  1207.6  3911.2 

6543.2  1025.8  1249.5  3984.9 

38 

40 

64G3.5  998.2  1209.0  3913.6 

6545.9  1026.7  1250.9  3987.4 

40 

42 

6466.3  999.2  1210.3  3916.1 

6548.6  1027.7  1252.3  3989.9 

42 

44 

6469.0  1000.1  1211.7  3918.5 

6551.4  1028.6  1253.7  3992.3 

44 

46 

6471.8  1001.0  1213.1  3921.0 

6554.1  1029.6  1255.1  3994.8 

46 

48 

6474.5  1002.0  1214.5  3923.4 

6556.9  1030.5  1256.5  3997.3 

48 

50 

6477.3  1002.9  1215.9  3925.9 

6559.6  1031.5  1257.9  3999.8 

50 

52 

6480.1  1003.8   217.3  3928.3 

6562.3  1032.4  1259.3  4002.2 

52 

51 

6482.8  1001.8   218.7  3930.8 

6565.1  1033.4  1260.7  4004.7 

54 

56 

6485.6  1005.7   220.1  3933.2 

6567.8  1034.3  1262.1  4007.2 

56 

58 

6488.3  1006.7   221.5  3935.7 

6570.6  1035.3  1263.5  4009.7 

58 

60 

6491.1  1007.7   222.9  3938.1 

6573.3  1036.3  1265.0  4012.1 

60 

228       FUNCTIONS  OF  A  ONE-DEGREE  CURVE 


70° 

71° 

L.  C.    M.    E.     T. 

L.  C    M.    E.     T. 

f 

0 

6573.3  1036.3  1265.0  4012.1 

6654.9  1065.1  1308.4  4087.1 

0 

2 

6576.0  1037.3  1266.4  4014.6 

6657  .6  1066.1  1309.9  4089.7 

2 

4 

6578.7  1038.2  1267.9  4017.1 

6660.3  1067.0  1311.3  4092.2 

4 

6 

6581.5  1039.2  1269.3  4019.6 

6663.0  1068.0  1312.8  4094!7 

6 

8 

6584.2  1040.1  1270.8  4022.1 

6665.7  1068.9  1314.2  4097.2 

8 

10 

65C6.9  1041.1  1272.2  4024.6 

6668.4  1069.9  1315.7  4099.8 

10 

18 

6589.6  104v!.l  1273.6  4027.1 

6671.1  1070.9  1317.2  4102.3 

12 

14 

6592.3  1043.0  1275.1  4029.6 

6673.8  1071.9  1318.6  4104.8 

14 

16 

6595.1  1044.0  1276.5  4032.1 

6676.6  1072.9  1320.1  4107.3 

16 

18 

6597.8  1044.9  1278.0  4034.6 

6679.3  1073.8  1321.5  4109.8 

18 

20 

6600.5  1045.9  1279.4  4037.1 

6682.0  1074.8  1323.0  4112.4 

20 

22 

6603.2  1046.9  1280.8  4039.6 

6684,7  1075.8  13'.>4.4  4114.9 

22 

24 

6605.9  1047.8  1282.3  4042.1 

6687  4  1076.8  1325.9  4117.4 

24 

26 

6608.7  1048.8  1283.7  404-1.6 

6690.1  1077.7  1327.4  4119.9 

26 

28 

6611.4  1049.7  1285.2  4047.1 

6692.8  1078.7  1328.9  4122.4 

28 

30 

6614.1  1050.7  1286.6  4049.6 

6695.5  1079.7  1330.4  4125.0 

30 

32 

6616.8  1051.7  1288.0  4052.1 

6693.2  1080.7  1331.8  4127.5 

32 

34 

6619.5  1052.6  1289.5  4054.6 

6700.9  1081.6  1333.3  4130.1 

34 

36 

6622.3  1053.6  1290.9  4057.1 

6703.6  1082.6  1334.8  4132.6 

36 

38 

6625.0  1054.5  1292.4  4059.6 

670(3.3  1083.6  1336.3  4135.1 

38 

40 

6627.7  1055.5  1293.8  4062.1 

6709.0  1084.5  1337.8  4137.7 

40 

42 

6630.4  1056.5  1295.3  4064.6 

6711.7  1085.5  1339.2  4140.2 

42 

44 

6633.1  1057.4  1296.7  4067.1 

6714.4  1086.5  1340.7  4142.7 

44 

46 

6635.9  1058  4  1298.2  4069.6 

6717.2  1087.5  1342.2  4145.  3 

46 

48 

6638.6  1059.3  1299.6  4072.1 

6719.9  1088.4  1343.7  4147.8 

48 

50 

6641.3  1060.3  1301.1  4074.6 

6722.6  1089.4  1345.2  4150.4 

50 

52 

6644.  C  1061.3  1302.6  4077.1 

6725.3  1090.4  1346.7  4152.9 

52 

£4 

6646.7  1002.2  1304.0  4079.6 

6728.0  1091.3  1348.2  4155.4 

51 

56 

6649.5  1063.2  1305.5  4082.1 

6730.7  1092.3  1349.7  4158.0 

56 

58 

6652.2  1064.1  1306.9  4084.6 

6733.4  1093.3  1351.2  4160.5 

58 

60 

6654.9  1065.1  1308.4  4087.1 

6736.1  1094.3  1352.7  4163.1 

60 

72° 

73° 

L.  C.    M.     E.    T. 

L.  C.    M.     E.    T. 

' 

0 

6736.1  1094.3  1352.7  4163.1 

6816.6  1123.9  1398.1  4240.0 

0 

2 

6738.8  1095  2  1354.2  4165.6 

6819.3  1124.8  1399.6  4242.6 

2 

4 

6741.5  1096.2  1355.7  4168.2 

6821.9  1125.8  1401.2  4245.1 

4 

6 

6744.1  1097.2  1357.2  4170.7 

68-J4.6  1126.8  1402.7  4247.7 

6 

8 

6746.8  1098.2  1358.7  4173.3 

6827.3  1127.8  1404.2  4250.3 

8 

10 

6749.5  1099.2  1360.2  4175.8 

6830.0  1128.8  1405.8  4252.9 

10 

12 

6752  2  1100.1  1361.7  4178.4 

6832.6  1129.8  1407.3  4255.5 

12 

14 

6754.9  1101.1  1363.2  4181.0 

6835.3  1130.8  1408.8  4258.1 

14 

16 

6757.6  1102.1  1364.7  4183.5 

6838.0  1131.8  1410.4  4260.7 

16 

18 

6760.2  1103.1  1366.2  4186.1 

6840.7  1132.8  1411.9  4263.2 

18 

20 

6762.9  1104.1  1367.7  4188.6 

6843.3  1133.8  1413.5  4265.8 

20 

22 

6765.6  1105.1  1369.2  4191.2 

684G.O  1134.8  1415.1  4268.4 

22 

24 

6768.3  1106.0  1370.7  4193.7 

6848.7  1135.8  1416.6  4271.0 

24 

26 

6771.0  1107.0  1372.2  4196.3 

6851.3  1136.8  1418.2  4273.6 

26 

28 

6773.7  1108.0  1373.7  4198.8 

6854.0  1137.8  1419.7  4276.2 

28 

30 

6776.3  1109.0  1375.2  4201.4 

6856.7  1138.8  1421.3  4278.8 

30 

32 

6779.0  1109.9  1376.7  4204.0 

6859.4  1139.8  1422.9  4281.4 

32 

34 

6781.7  1110.9  1378.2  4206.5 

6862.0  1140.8  1424.4  4284.0 

34 

36 

6784.4  1111.9  1379.7  4209.1 

6864.7  1141.8  1426.0  4286.6 

36 

38 

6787.1  1112.9  1381.2  4211.7 

6867.4  1142.8  1427.5  4289.2 

38 

40 

6789.8  1113.9  1382.8  4214.3 

6870.1  1143.8  1429.1  4291.8 

40 

42 

6792.4  1114.9  1384.3  4216.8 

6872.7  1144.8  1430.7  4294.4 

42 

44 

6795.1  1115.9  1385.8  4219.4 

6875.4  1145.8  1432.2  4297.0 

44 

46 

6797.8  1116.9  1387.4  4222.0 

6878.1  1146.8  1433.8  4299.6 

46 

48 

6800.5  1117.9  1388.9  4224.5 

6880.8  1147.8  1435.3  4302.2 

48 

50 

6803.2  1118.9  1390.4  4227.1 

6883.4  1148  8  1436.9  4304.8 

50 

52 

6805.9  1119.9  1392.0  4229.7 

6886.1  1149.8  1438.5  4307.4 

52 

54 

6808.5  1120.9  1393.5  4232.3 

6888.8  1150.8  1440.0  4310.0 

54 

56 

6811.2  1121.9  1395.0  4234.8 

6891.4  1151.8  1441.6  4312.6 

56 

68 

6813.9  1122.9  1396.6  4237.4 

6894.1  1152.8  1443.1  4315.2 

58 

60 

6816.6  1123.9  1398.1  4240.0 

6896.8  1153.8  1444.7  4317.8 

60 

FUNCTIONS  OF  A  ONE-DEGREE  CURVE       229 


740 

76° 

L.  C.    M.    E.     T. 

L.  C.    M.    E.     T. 

0 

6890.8  1153.8  1444.7  4317.8 

6976.4  1184.1  1492.5  4396.7 

0 

2 

6899.4  1154.8  1446.2  4320.5 

6979.0  1185.1  1494.1  4399.4 

2 

4 

6902.1  1155.8  1447.8  4323.1 

6981.7  1186.1  1495.7  4402.1 

4 

6 

6904.8  1156.8  1449.4  4325.7 

6984.3  1187.1  1497.3  4404.7 

6 

8 

0907.4  1157.8  1451.0  4328.3 

6986.9  1188.1  1499.0  4407.4 

8 

10 

6910.1  1158.8  1452.6  4330.9 

6989.6  1189.2  1500.6  4410.0 

10 

12 

6912.7  1159.8  1454.1  4333.6 

699-,'.  2  1190.2  1502.2  4412.7 

12 

14 

6915.4  1160.8  1455.7  4336.2 

6994.9  1191.2  1503.8  4415.3 

14 

16 

6918.0  1161.8  1457.3  4338.8 

6997.5  1192.2  1505.4  4418.0 

16 

18 

6920.7  1162.8  1458.9  4341.4 

7000.1  1193.2  1507.0  4420.7 

18 

20 

6923.3  1163.9  1460.5  4344.0 

7002.8  1194.3  1508.7  4423.3 

20 

22 

C926.0  1164.9  1462.0  4346.7 

7005.4  1195.3  1510.3  4426.0 

22 

24 

6928.6  1165.9  1463.6  4349.3 

7008.0  1196.3  1512.0  4428.6 

24 

26 

6931.3  1166.9  1465.2  4351.9 

7010.7  1197.3  1513.6  4431.3 

26 

28 

6933.9  1167.9  1466.8  4354.5 

7013.3  1198.3  1515.3  4434.0 

28 

30 

6936.6  1168.9  1468.4  4357.1 

7015.9  1199.4  1516.9  4436.6 

30 

32 

6939.2  1169.9  1469.9  4359.8 

7018.6  1200.4  1518.5  4439.3 

32 

34 

6941.9  1170.9  1471.5  4362.4 

7021.2  1201.4  1520.2  4442.0 

34 

3« 

6944.6  1171.9  1473.1  4365.1 

7023.9  1-202.4  1521.8  4444.6 

36 

38 

6947.2  1172.9  1474.7  4367.7 

7026.5  1203.4  1523.5  4447.3 

33 

40 

6949.9  1174.0  1476.4  4370.3 

7029.1  3204.5  1525.1  4450.0 

40 

42 

6952.5  1175.0  1478.0  4373.0 

7031.8  1205.5  1526.7  4452.  7 

42 

44 

6955.2  1176.0  1479.6  4375.6 

7034.4  1206.5  1528.4  4455.3 

44 

46 

6957.8  1177.0  1481.2  4378.3 

7037.0  1207.5  1530.0  4458.0 

46 

48 

0960.5  1178.0  1-182.8  4380.9 

7039.7  1208.5  1531.7  4460.7 

48 

50 

6963.1  1179.0  1484.4  4383.5 

7042.3  1209.6  1533.3  4463.4 

50 

5:2 

6965.8  1180.0  1486.0  4386.2 

7045.0  1210.6  1534.9  4466.0 

52 

54 

6968.4  1181.0  1487.7  4388.8 

7047.6  1211.6  1536.6  4468.7 

54 

56 

6971.1  1182.0  1489.3  4391.5 

7050.2  1212.6  1538.2  4471.4 

56 

58 

6973.7  1183.0  1490.9  4394.1 

705-2.9  1213.6  1539.9  4474.1 

58 

60 

6976.4  118-1.1  149:2.5  439(5.7 

7055.5  1211.7  1541.5  4476.7 

60 

!  , 

76° 

JJO 

L.  C.    M.    E.     T. 

L.  C.    M    E.     T. 

0 

7055.5  1214.7  1541.5  4476.7 

7134.0  1245.6  1591.7  4557.8 

0 

2 

7058.1  1215.7  1543.2  4479.4 

7136.6  1246.6  1593.4  4560.5 

2 

4 

7060.7  1216.7  1544.9  4482.1 

7139.2  1247.7  1595.1  4563.3 

4 

6 

7063.3  1217.8  1546.5  4484.8 

7141.8  124S.7  1596.8  4566.0 

6 

8 

7066.0  1218.8  1548.2  4487.5 

7144.4  1249.8  1598.5  4568.7 

8 

10 

7068.6  1219.8  1549.9  4490.2 

7147.0  1250.8  1600.2  4571.5 

10 

12 

7071.2  1220.9  1551.5  4492.9 

7149.6  1251.8  1601.9  4574.2 

12 

14 

7073.8  1221.9  1553.2  4495.6 

7152.2  1252.9  1603.6  4576.9 

14 

16 

7076.4  1222.9  1554.9  4498.3 

7154.8  1253  9  1605  3  4579.7 

16 

18 

7079.0  1224.0  1556.5  4501.0 

7157.4  12550  1607.0  4582.4 

18 

20 

7081.7  1225.0  1558.2  4503.7 

7160.0  1256.0  1608.7  4585.1 

20 

22 

7084.3  1226.0  1559.9  4506.8 

7162.6  1257.0  1610.4  4587.9 

22 

24 

7086.9  1227.1  1561.5  4509.0 

7165.2  1258.1  1612.1  4590.6 

24 

26 

7089.5  1228.1  1563.2  4511.7 

7167.8  1259.1  1613.8  4593.3 

26 

28 

7092.1  1229.1  1564  9  4514.4 

7170.4  1260.2  1615.5  4596.0 

28 

30 

7094.7  1230.2  1566.5  4517.1 

7173.0  1261  2  16'17.3  4598  8 

30 

32 

7097.4  1231.2  1568.2  4519.8 

7175.6  12622  1619.0  4601.5 

32 

31 

7100.0  1232.2  1569.9  4522.5 

71782  1263.3  1620.7  4604.3 

34 

36 

7102.6  1233.3  1571.5  4525.3 

7180  8  12S4.3  1622.4  4607  0 

36 

38 

7105.2  1234.3  1573.2  4528.0 

7183.4  1265.4  1624.1  46098 

38 

40 

7107.8  1235.3  1574.8  4530.7 

7186.0  1266.4  1625.9  4612.5 

40 

42 

7110.4  1236.4  1576.4  4533.4 

7188  6  1267.4  1627.6  4615.3 

42 

44 

7113.1  1237.4  1578.1  4536.1 

7191.2  1268.5  1629.3  4618.0 

44 

46 

7115.7  1238.4  1579.8  4538.8 

7193.8  1269.5  1631.0  4620.8 

46 

48 

7118.3  12395  1581.5  4541.5 

7196.4  1270  6  1632.7  4623.5 

48 

50 

7120.9  1240.5  1583.2  4544.2 

7199.0  1271.6  1634  5  4626.3 

50 

52 

7123.5  1241.5  1584.9  4547.0 

7201.6  1272.7  16362  4629.0 

52 

54 

7126.1  1242.6  1586.6  4549.7 

72042  1273.7  1687.9  4631.8 

54 

56 

7128.8  1243.6  1588.3  4552.4 

7206.8  1274.8  1639.6  4634.5 

56 

58 

7131.4  1244.6  15900  4555.1 

7209.4  1275.8  1641.3  4637.3 

58 

6C 

7134.0  1245.6  1591.7  4557.8 

7212.0  1276.9  1643.1  4640.0 

60 

230       FUNCTIONS  OF  A  ONE-DEGREE   CURVE 


78° 

79° 

L.  C.   M.    E.    T. 

L.  C.    M.     E.    T. 

0 

7212.0  1276.9  1643.1  4640.0 

7289.5  1308.5  1696.0  4723.4 

0 

2 

7214.6  1278.0  1644.8  4642.8 

7292.1  1309.5  1697.7  4726.2 

2 

4 

7217.2  1279.0  1646.6  4645.6 

7294.6  1310.6  1699.5  4729.0 

4 

6 

7219.7  1280.1  1648.3  4648.3 

7297.2  1311.7  1701.3  4731.8 

6 

8 

7222.3  1281.1  1650.1  4651.1 

7299.7  1312.7  1703.1  4734.7 

8 

10 

7224.9  1282.2  1651.8  4653.9 

7302.3  1313.8  1704.9  4737.5 

10 

12 

7227.5  1283.2  1653.6  4656.7 

7304.9  1314.9  1706.6  4740.3 

12 

14 

7230.1  1284.3  1655.3  4659.4 

7307.4  1315.9  1708.4  4743.1 

14 

16 

7232.7  1285.3  1657.1  4662.2 

7310.0  1317.0  1710.2  4745.9 

16 

18 

7235.2  1286.4  1658.8  4665.0 

7312.6  1318.1  1712.0  4748.7 

18 

20 

7237.8  1287.4  1660.6  4667.7 

7315.1  1319.1  1713.8  4751.5 

20 

22 

7240.4  1288.5  5662.3  4670.5 

7317.7  1320.2  1715.6  4751.3 

22 

24 

7243.0  1289.5  1664.1  4673.3 

7320.3  1321.3  1717.4  4757.1 

24 

26 

7245.6  1290.6  1665.8  4676.0 

7322.8  1322.3  1719.2  4760.0 

26 

28 

7248.2  1291.6  1667.6  4678.8 

7325.4  1323.4  1721.0  4762.8 

28 

30 

7250.7  1292.7  1G69.3  4681.6 

7327.9  1324.5  1722.8  4765.6 

30 

82 

7253.3  1293.7  1671.1  4684.4 

7330.5  1325.5  1724.6  4768.4 

32 

34 

7255.9  1294.8  1672.8  4687.2 

7333.1  13-J6.6  1726.4  4771.2 

34 

36 

7258.5  1295.8  1674.6  4689.9 

7335.6  1327.7  1728.2  4774.1 

36 

38 

7261.1  1296.9  1676.3  4692.7 

7338.2  1328.  7  17*0.0  4776.9 

38 

40 

7263.7  1297.9  1678.2  4695.5 

7340.8  1329.8  1731.9  4779.7 

40 

42 

7266.2  1299.0  1679.9  4698.3 

7343.3  1330.8  173S.7  4782.6 

4'^ 

44 

7268.8  1300.0  1681.7  4701.1 

7345.9  1331.9  1735.5  4785.4 

44 

46 

7271.4  1301.1  1683.5  4703.9 

7348.4  1333.0  1737.3  4788.2 

46 

48 

7274.0  1302.1  1685.3  4706.7 

7351.0  1334.1  1739.1  4791.0 

48 

50 

7276.6  1303.2  1687.1  4709.5 

7353.6  1335.2  1740.9  4793.9 

50 

52 

7279.2  1304.2  1688.8  4712.2 

7356.1  1336.2  1742.7  4796.7 

52 

54 

7281.7  1305.3  1690.6  4715.0 

7358.7  1337.3  1744.5  4799.5 

54 

56 

7284.3  1306.3  1692.4  4717.8 

7361.3  1338.4  1746.3  4802.4 

56 

58 

7286.9  1307.4  1694.2  4720.6 

7363.8  1339.5  1748.1  4805.  -2 

58 

60 

7289.5  1308.5  1696.0  4723.4 

7366.4  1340.6  1750.0  4808.0 

60 

80° 

81° 

L.  C.   M.     E.     T. 

L.  C.   M.     E.     T 

0 

7366.4  1340.6  1750  0  4808.0 

7442.7  1372.8  3805.5  4893.9 

0 

2 

7368  9  1341.7  1751.8  4810.9 

7445.2  1373.9  1807.3  4896.8 

2 

4 

7371.5  1342.7  1753.7  4813.7 

7447.7  1375.0  1809.2  4899.7 

4 

6 

7374.0  1343.8  1755.5  4816.6 

7450.3  1376.1  1811.1  4902.6 

6 

8 

7376.6  1344.9  1757.4  4819.4 

74M.8  1377.1  1813.0  4905.4 

8 

10 

7379.1  1346.0  1759.2  4822.3 

7455.3  1378.2  1814.9  4908.3 

10 

12 

7381.7  1347.0  1761.0  4825.1 

7457.8  1379.3  1816  8  4911.2 

12 

14 

7384.2  1348.1  1762.9  4828.0 

7460.4  1380.4  1818.6  4914.1 

14 

16 

7386.7  1349.2  1764.7  4830.  H 

7462.9  1381.4  1820.5  4917.0 

16 

18 

7389.3  1350.3  1766.6  4833.7 

7465.4  1382.5  1822.4  4919.9 

18 

20 

7391.8  1351.3  1768.4  4836.5 

7467.9  1383.6  1824.2  4922.8 

20 

22 

7394.4  1352.4  1770.2  4839.4 

7470.4  1384.7  1826.1  4925.7 

22 

24 

7396.9  1353.5  1772.1  4842.2 

7473.0  1385.7  1828.0  4928.6 

24 

26 

7399.5  1354.6  1773.9  4845.1 

7475.5  1386.8  1829.9  4931.5 

26 

28 

7402.0  1355.6  1775.8  4847.9 

7478.0  1387.9  1831.8  4934.4 

28 

30 

7404.5  1356.7  1777.6  4850.8 

7480.5  1389.0  1833.7  4937.2 

30 

32 

7407.1  1357.8  1779.4  4853.7 

7483.1  1390.1  1835.6  4940.2 

82 

34 

7409.6  1358.9  1781.3  4856.5 

7485.6  1391.2  1837.5  4943.1 

34 

36 

7412.2  1359.9  1783.1  4859.4 

7488.  J  1392.3  1839.4  4946.0 

36 

38 

7414.7  1361.0  1785.0  4862.3 

7490.6  1393.4  1841.3  4948.9 

38 

40 

7417.3  1362.1  1786.8  4865.1 

7493.2  1394.5  1843.2  4951.8 

40 

42 

7419  8  1363  2  1788.6  4868.0 

7495.7  1395.6  1845.1  4954.7 

42 

44 

7422.3  1364.2  1790.5  4870.9 

7498.2  1396.7  1847.0  4957.6 

44 

46 

7424.9  1365.3  1792.4  4873.8 

7500.7  1397.8  1848.9  4960.6 

46 

48 

7427.4  1366.4  1794  3  4876.6 

7503.3  1398.9  1850.8  4963.5 

48 

50 

7430.0  1367.5  1796.2  4879.5 

7505.8  1400.0  1852  7  4966.4 

50 

52 

7432.5  1368.5  1798.0  4882.4 

7508.3  1401.1  1854  6  4969.3 

52 

54 

7435.1  1369.6  1799.9  4885  3 

7510.8  1402.2  1856  5  4972.2 

54 

56 

7437.6  1370  7  1801.8  4888.1 

7513.3  1403.3  185S.4  4975.1 

56 

58 

7440.1  1371.8  1803.7  4891.0 

7515.9  1404  4  1860.3  4978.0 

58 

60 

7442.7  1372.8  1805.5  4893.9 

7518.4  1405.5  1862  3  4981.  C 

SO 

FUNCTIONS  OF   A  ONE-DEGREE   CURVE       231 


82° 

83° 

L.  C.   M.     E.     T 

L.  C.   M.     E.     T. 

0 

7513.4  1405.5  1862.3  4981.0 

7593.6  1438.5  1920.6  5069.4 

0 

2 

7520.9  1406.6  1864.2  4983.9 

7596.1  1439.6  1922.6  5072.4 

2 

4 

7523.4  1407.7  1866.1  4986.8 

7598.6  1440.7  1924.6  5075.4 

4 

6 

7525.9  1408  8  1868.1  4989.8 

7601.1  1441.8  1926.5  5078.4 

6 

8 

7528.4  1409.9  1870.0  4992.7 

7603.6  1442.9  1928.5  5081.4 

8 

10 

7530.9  1411.0  1871  9  4995.7 

7606.0  1444.0  1930.5  5084.4 

10 

12 

7533.4  1412.1  1873  9  4998.6 

7608.5  1445.1  1932.4  5087.3 

12 

14 

7535.9  1413.2  1875.8  5001.5 

7611.0  1446.2  1934.4  5090.3 

14 

16 

7538.5  1414.3  1877.7  5004.5 

7613.5  1447.3  1936.4  5093.3 

16 

18 

7541.0  1415.4  1879.7  5007.4 

7616.0  1448.4  1938.4  5096.  3 

18 

20 

7543.5  1416.5  1881.6  5010.3 

7618.5  1449.6  1940.4  5099.3 

20 

22 

7546.0  1417.6  1883.5  5013.3 

7621.0  1450.7  1942.4  5102.3 

22 

24 

7548.5  1418.7  1885.5  5016.2 

7623.5  1451.8  1944.4  5105.2 

24 

26 

7551.0  1419.8  1887.4  5019.2 

7626.0  1452.9  1946.4  5108.2 

26 

28 

7553.5  1420.9  1889.3  5022.1 

7628.5  1454.0  1948.4  5111.2 

28 

30 

7556.0  1422.0  1891.3  5025.0 

7630.9  1455.1  1950.4  5114  2 

30 

32 

7558.5  1423.1  1893.2  5028.0 

7633.4  1456.2  1952.4  5117.2 

32 

34 

7501.0  1424.2  1895.1  5031.0 

7G35.9  1457.3  1954.4  5120.2 

34 

36 

7563.5  14-25.3  1897.1  5033.9 

7638.4  1458.4  1956.4  5123.2 

36 

38 

7566.0  1426.4  1899.0  5036.9 

7640.9  1459.5  1958.4  5126.2 

38 

40 

7568.5  1427.5  1901.0  5039.8 

7643.4  1460.7  1960.4  5129.2 

40 

42 

7571.0  1428.6  1902.9  5042.8 

7645.9  1461.8  1962.4  5132.2 

42 

44 

7573.5  1429.7  1904.9  5045.8 

7648.4  1462.9  1964.4  5135.2 

44 

46 

7576.1  1430.8  1906.9  5048.7 

7650.9  1464.0  1966.4  5138.2 

46 

48 

7578.6  H31.9  1908.8  5051.7 

7653.4  1465.1  1968.4  5141.2 

48 

50 

7581.1  1433.0  1910.8  5054.6 

7655.8  1466.2  1970.4  5144.3 

50 

52 

7583.6  1434.1  1912.8  5057.6 

7658.3  1467.3  1972.4  5147.3 

52 

54 

7586.1  1435.2  1914.7  5060.6 

7660.8  1468.4  1974.4  5150.3 

54 

56 

7588.6  1436.3  1916.7  5063.5 

7663.3  14C9.5  1976.4  5153.3 

56 

58 

7591.1  1437.4  1918.7  5066.5 

7665.8  1470.6  1978.4  5156.3 

58 

60 

7593.6  1438.5  1920.6  5069.4 

7668.3  1471.8  1980.5  5159.3 

60 

84-° 

85° 

/ 

L.  C.    M.    E.     T. 

L.  C.    M.    E.     1 

0 

7668.3  1471.8  1980.5  5159.3 

7742.4  1505.4  2041.8  5250.6 

0 

2 

7670.8  1472.9  1982.5  5162.3 

7744.8  1506.5  2043.9  5253.6 

2 

4 

7673.2  1474.0  1984.5  5165.3 

7747.3  1507.6  2046.0  5256.7 

4 

6 

7675.7  1475.1  1986.6  5168.4 

7749.7  1508.8  2048.0  5259.8 

6 

8 

7(378.2  1476.2  1988.6  5171.4 

7752.2  1509.9  2050.1  5262.9 

8 

10 

7680.6  1477.4  1990.6  5174.4 

7754.6  1511.0  2052.2  5266  0 

10 

12 

7683.1  1478.5  1992.7  5177.5 

7757.1  1512.2  2054.2  5269.0 

12 

14 

7685  6  1479.6  1994.7  5180.5 

77595  1513.3  2056.3  5272.1 

14 

16 

7688.1  1480.7  1990.7  5183.5 

7762.0  1514.4  2058.4  5275.2 

16 

18 

7690.5  1481.8  1998.8  5186.6 

7764  4  1515.6  2060.5  5278.3 

18 

20 

7693.0  14&3.0  2000.8  5189.6 

7766.9  1516.7  2062.6  5281.4 

20 

22 

7695.5  1484  1  2002.8  5192.6 

7769.3  1517.8  2064.7  5284.4 

22 

24 

769r.9  1485.2  2004.9  5195.6 

7771.8  1519.0  2066.8  5287.5 

24 

26 

7700.4  1486.3  2006.9  5198.7 

7774.2  1520.1  2068.9  5290.6 

26 

ass 

770'2.9  1487.4  2008.9  5201.7 

7776.7  1521.2  2071.0  5293.7 

28 

30 

7705.3  1488  6  2011.0  5204.7 

7779.1  1522.4  2073.  f  5296.7 

30 

32 

7707.8  1489.7  2013.0  5207.8 

7781.5  1523.5  2075  2  5299  8 

32 

34 

7710.3  1490.8  2015.0  5210  8 

7784.0  1524.6  2077  3  5302.9 

34 

36 

7712.8  1491.9  2017.0  5213.9 

7786.4  1525.8  2079.4  5306.1 

86 

38 

7715.2  1493.0  2019.1  5216.9 

7788.9  1526.9  2081.5  5309.2 

38 

40 

7717.7  1494.2  2021.2  5220.0 

7791.3  1528.0  2083.7  5312.3 

40 

42 

77202  1495.3  2023.2  5223.1 

7793.8  1529.2  2085.8  5315.4 

42 

44 

7722  6  1496.4  2025.3  5226.1 

7796  2  1530.3  2087.9  5318.5 

44 

46 

7725.1  1497-5  2027.4  5229.2 

77987  1531.4  2090.0  5321.6 

46 

48 

7727.6  1498.6  2029.4  5232.2 

7801.1  1532.6  2092.1  5324.7 

48 

50 

7730.0  14998  2031.5  5235.3 

7803.6  1533.7  2094.2  5327.8 

50 

52 

7732  5  1500.9  2033.6  5238.3 

7806.0  1534  8  2096.3  5330  9 

52 

54 

7735.0  1502.0  2035.6  5241  4 

7808.5  1536.0  2098.4  5334.0 

54 

56 

7737  5  1503.1  2037.7  5244  5 

7810.9  1537.1  2100.6  5337.1 

56 

58 

7739  9  1504.2  2039.8  5247.5 

7813  4  1538.2  2102.7  5340.2 

58 

60 

7742.4  15054  2041.8  5250.6 

7815.8  1539.3  2104.8  5343.3 

60 

232       FUNCTIONS  OF  A  ONE-DEGREE   CURVE 


86° 

87° 

L.  C.    M.    E.    ,T. 

L.  C.   M.    E.    T. 

/ 

0 

7815.8  1639.3  2104.8  5343.3 

7888.5  1573.6  2169.5  5437.5 

0 

2 

7818.2  1540.4  2106.9  5346.4 

7890.9  1574.8  2171.6  5440.7 

2 

4 

7820.6  1541.6  2109.1  5349.5 

7893.3  1575.9  2173.8  5443.9 

4 

6 

7823.1  1542.7  2111.2  5352.7 

7895.7  1577.1  2176.0  5447.1 

6 

8 

7825.5  1543.9  2113.4  5355.8 

7898.1  1578.2  2178.2  5450.3 

8 

10 

7827.9  1545.0  2115.5  5358.9 

7900.5  1579.4  2180.4  5453.4 

10 

12 

7830.3  1546.1  2117.6  5362.0 

7903.0  1580.5  2182.5  5456.6 

12 

14 

7832.8  1547.3  2119.8  5365.2 

7905.4  1581.7  2184.7  5459.8 

14 

16 

7835.2  1548.4  2121.9  5368.3 

7907.8  1582.9  2186.9  5463.0 

16 

18 

7837.6  1549.6  2124.1  5371.4 

7910.2  1584.0  2189.1  5466.2 

18 

20 

7840.0  1550.7  2126.2  5374.6 

7912.6  1585.1  2191.3  5469.4 

20 

22 

7812.4  1551.8  2128.3  5377.7 

7915.0  1586.3  2193.5  5472.5 

22 

24 

7844.9  1553.0  2130.5  5980.8 

7917.4  1587.4  2195.7  5475.7 

24 

26 

7847.3  1554.1  2132.6  5383.9 

7919.8  1588.6  2197.9  5478.9 

26 

28 

7849.7  1555.3  2134.8  5387.1 

7922.2  1589.7  2200.1  5482.1 

28 

30 

7852.1  1556.4  2136.9  5390.2 

7924.6  1590.9  2202.3  5485.3 

30 

32 

7854.6  1557.5  2139.0  5393.4 

7927.1  1592.0  2204.5  5488.5 

S2 

34 

7857.0  1558.7  2141.2  5396.5 

7929.5  1593.2  2206.8  5491.7 

34 

36 

7859.4  1559.8  2143.3  5399.7 

7931.9  1594.3  2209.0  5494.9 

36 

38 

7861.8  1561.0  2145.5  5402.8 

7934.3  1595.5  2211.2  5498.1 

38 

40 

7864.3  1562.1  2147.7  5406.0 

7936.7  1596.6  2213.4  5501.3 

40 

42 

7866.7  1563.2  2149.8  5409.1 

7939.1  1597.8  2215.6  5504.5 

42 

44 

7869.1  1564.4  2152.0  5412.3 

7941.5  1598.9  2217.8  5507.7 

44 

46 

7871.5  1565.5  2154.2  5415.4 

7943.9  1600.1  2220.0  5510.9 

46 

48 

7874.0  1566.7  2156.4  5418.6 

7946.3  1601.2  2222.3  5514.1 

48 

50 

7876.4  1567.»  2158.6  5421.8 

7948.7  1602.4  2224.5  5517.3 

50 

52 

7878.8  1568.9  2160.7  5424.9 

7951.2  1603.5  2226.7  5520.5 

52 

54 

7881.2  1570.1  2162.9  5428.1 

7953.6  1604.7  2228.9  5523.7 

54 

56 

7883.6  1571.2  2165.1  5431.2 

7956.0  1605.8  2231.1  5526.9 

56 

58 

7886.1  1572.4  2167.3  5434.4 

7958.4  1607.0  2233.3  5580.1 

58 

60 

7888.5  1573.6  2169.5  5437.5 

7960.8  1608.2  2235.6  5533.3 

60 

88° 

89° 

L.  C.    M.    E.     T. 

L.  C.   M.    E.     T. 

0 

7960.8  1608.2  2235.6  5533.3 

8032.4  1643.0  2303.6  5630.8 

0 

2 

7963.2  1609.4  2237.8  5536.6 

8034.8  1644.1  2305.9  5634.1 

2 

4 

7965.6  1610.5  2240.1  5539.8 

8037.1  1645.3  2308.2  5637.4 

4 

6 

7968.0  1611.7  2242.3  5543.1 

8039.5  1646.5  2310.5  5640.7 

6 

8 

7970.3  1612.8  2344.6  5546.3 

8041.9  1647.7  2312.8  5644.0 

8 

10 

7972.7  1614.0  2246.8  5549.5 

8044.2  1648.9  2315.1  5647.3 

10 

1? 

7975.1  1615.2  2249.1  5552.8 

8046.6  1650.0  2317.4  5650.6 

12 

14 

7977.5  1616.3  2251.3  5556.0 

8049.0  1651.2  2319.7  5653.9 

14 

16 

7979.9  1617.5  2253.6  5559.2 

8051.4  1652.4  2322.0  5657.1 

16 

18 

7982.3  1618.6  2255.8  5562.5 

8053.7  1653.6  2324.3  5660.4 

18 

20 

7984.7  1619.8  2258.1  5565.7 

8056.1  1654.8  2326.7  5663.7 

20 

22 

7987.1  1621.0  2260.4  5568.9 

8058.5  1655.9  2329.0  5667.0 

22 

24 

7989.4  1622.1  2262.7  5572.2 

8060.8  1657.1  2331.3  5670.3 

24 

26 

7991.8  1623.3  2264.9  5575.4 

8063.2  1658.3  2333.7  5673.6 

26 

28 

7994.2  1624.4  2267.2  5578.6 

8065.6  1659.5  2336.0  5676.9 

28 

30 

7996.6  1625.6  2269.5  5581.9 

8067.9  1660.7  2338.3  5680.2 

30 

32 

7999.0  1626.8  2271.7  5585.1 

8070.3  1661.8  2340.7  5683.5 

32 

34 

8001.4  1627.9  2273.9  5588.4 

8073.7  1663.0  2343.0  5686.8 

34 

36 

8003.8  1629.1  2276.2  5591.7 

8075.1  1664.2  2345.3  5690.2 

36 

38 

8006.1  1630.2  2278.5  5594.9 

8077.4  1665.4  2347.7  5693.5 

38 

40 

8008.5  1631.4  2080.8  5598.2 

8079.8  1666.6  2350.0  5696.8 

40 

42 

8010.9  1632.6  2283.0  5601.4 

8082.2  1667.7  2352.3  5700.1 

42 

44 

8013.3  1633.7  2285.3  5604.7 

8084.5  1668.8  2354.7  5703.4 

44 

46 

8015.7  1634.9  2^87.6  5608.0 

8086.9  1670.0  2357.0  5706.8 

46 

48 

8018.1  1636.0  2289  9  5611.2 

8089.3  1671.2  2359.3  5710.1 

48 

50 

8020.5  1637.2  2292.2  5614.5 

8091.6  1672.4  2361.7  5713.4 

50 

52 

8022.9  1638.4  2294.4  5617.8 

8094.0  1673.5  2364.0  5716.7 

52 

54 

8025.2  1639.5  2296.7  5621.0 

8096.4  1674.7  2366.3  5720.0 

54 

56 

8027.6  1640.7  2299.0  5624.3 

8098.8  1675.9  2368.7  5723.4 

56 

58 

8030.0  1641.8  2301.3  5627.5 

8101.1  1677.1  2371.0  5726.7 

58 

60 

8032.4  1643.0  2303.6  5630.8 

8103.5  1678.3  2373.4  5730.0 

60 

FUNCTIONS  OF  A  ONE-DEGREE   CURVE       233 


90° 

91° 

L.  C.    M.    E.     T. 

L.  C.   M.    E.     T. 

/ 

0 

8103.5  1678.3  2373.4  5730.0 

8173.9  1713.8  2445.1  5830.9 

0 

2 

8105.8  1679.5  2375.8  5733.3 

8176.2  1715.0  2447.5  5834.3 

2 

4 

8108.2  1680.6  2378.2  5736.7 

8178.5  1716.2  2450.0  5837.7 

4 

6 

8110.5  1681.8  2380.5  5740.0 

8180.9  1717.4  2452.4  5841.1 

6 

8 

8112.9  1683.0  2382.9  5743.4 

8183.2  1718.6  2454.8  5844.5 

8 

10 

8115.2  1684.2  2385.3  5746.7 

8185.5  1719.7  2457.2  5847.9 

10 

12 

8117.6  1685.4  2387.6  5750.0 

8187.9  1720.9  2459.7  5851.3 

12 

14 

8119.9  1C86.5  2390.0  5753.4 

8190.2  1722.1  2462.1  5854.7 

14 

16 

8122.3  1687.7  2392.4  5756.7 

8192.5  1723.3  2464.5  5858.1 

16 

18 

8124.6  1688.9  2394.7  5760.1 

8194.8  1724.5  2467.0  5861.5 

18 

20 

8127.0  1690.1  2397.1  5763.4 

8197.2  1725.7  2469.4  5864.9 

20 

2-2 

81  -,'9.  3  1691.3  2399.5  5766.8 

8199.5  1726.9  2471.9  5868.3 

22 

24 

8131.7  1692.5  2401.9  5770.1 

8201.8  1728.1  2474.3  5871.8 

24 

20 

8134.0  1693.6  2404.3  5773.5 

8204  .2  1  729  .  3  2476  .  7  5875  .  2 

26 

28 

8136.4  1694.8  2406.6  5776.9 

8206.5   730.5  2479.2  5878.6 

28 

30 

8138.7  1696.0  2409.0  5780.2 

8208.8   731.7  2481.6  5882.0 

30 

32 

8141.1  1697.2  2411.4  5783.6 

8211.1   732.9  2484.1  5885.4 

32 

34 

8143.4  1698.  4  2413.8  5787.0 

8213.5   734.1  2486.5  5888.9 

34 

36 

8145.8  1699.6  2416.2  5790.3 

8215.8   735.3  2489.0  5892.3 

36 

38 

8148.1  1700.7  2418.6  5793.7 

8218.1   736.4  2491.5  5895.7 

38 

40 

8150.4  1701.9  2421.0  5797.1 

8220.4   737.6  2493.9  5899.2 

40 

4-2 

8152.8  1703.1  2423.4  5800.4 

8222.8   738.8  2496.4  5902.6 

42 

44 

8155.1  1704.3  2425.8  5803.8 

8225.1   740.0  2498.9  5906.0 

44 

46 

8157.5  1705.5  2428.2  5807.2 

8227.4   741.8  2501.3  5909.4 

46 

48 

8159.8  1706.7  2430.6  5810.6 

8229.7   742.4  2503.8  5912.9 

48 

50 

8162.2  1707.9  2433.0  5814.0 

8232.0   743.6  2506.3  5916.3 

50 

52 

8164.5  1709.0  2435.4  5817.3 

8234.3   744.8  2508.7  5919.8 

52 

54 

8166.8  1710.2  2437.9  5820.7 

8236.7   746.0  2511.2  5923.2 

54 

56 

8169.2  1711.4  2440.3  5824.1 

8239.0   747.2  2513.7  5926.7 

56 

58 

8171.5  1712.6  2442.7  5827.5 

8241.3  1748.4  2516.2  5930.1 

58 

60 

8173.9  1713.8  2445.1  5830.9 

8243.6  1749.6  2518.7  5933.6 

60 

92° 

93° 

L.  C    M.    E.     T. 

L.C.   M.    E.     T. 

0 

8243.6-  1749.6  2518.7  5933.6 

8312.8  1785.7  2594.2  6088.2 

0~ 

2 

8245.9  1750.8  2521.2  5937.0 

8315.1  1786.9  2596.8  6041  7 

2 

4 

8248.2  1752.0  2523.6  5940.5 

8317.4  1788.2  2599.3  6045.2 

4 

6 

8250.6  1753.2  2526.1  5944.0 

8319.7  1789.4  2601.9  6048.7 

6 

8 

8252.9  1754.4  2528.6  5947.4 

8322.0  1790,6  2604.4  6052.2 

8 

10 

8255.2  1755.6  2531.1  5950.9 

8324.3  1791.8  2(507.0  6055.8 

10 

12 

8257.5  1756.8  2533.6  5954.4 

8306.6  1793.0  2609.6  6059.3 

12 

14 

8259.8  1758.0  2536.1  5957.8 

8308.8  1794.2  2612.1  6062.8 

14 

16 

8262.2  1759.2  2538.6  5961.3 

8331.1  1795.4  2614.7  6066.4 

16 

18 

8264.5  1760.4  2541.1  5964.8 

8333.3  1796.6  2617.3  6069.9 

18 

20 

8266.8  1761.6  2543.6  5968.2 

8335.6  1797.8  2619.8  6073.4 

20 

82 

8269.1  1762.8  2546.1  5971.7 

8337.9  1799.1  2622.4  6077.0 

22 

24 

8271.4  1764.0  2548.6  5975.2 

8340.2  1800.3  2625.0  6080.5 

24 

26 

8273.7  1765.2  2551.2  5978.7 

8342.5  1801.5  2627.6  6084.1 

26 

28 

8276.0  1766.4  2553.7  5982.2 

8344.8  1802.7  2630.2  6087.6 

28 

30 

8278.3  1767.6  2556.2  5985.6 

8347.1  1803.9  2632.7  6091.2 

30 

32 

8280.6  1768.8  2558.7  5989.1 

8349.4  1805.1  2635.3  6094.7 

32 

34 

8282.9  1770.0  2561.2  5992.6 

8351.7  1806.3  2637.9  6098.3 

34 

36 

8285.2  1771.2  2563.8  5996.1 

8354.0  1807.6  2640.5  6101.8 

36 

38 

8287.5  1772.5  2566.3  5999.6 

8356.3  1808.8  2643.1  6105.4 

38 

40 

8289.8  1773.7  2568.8  6003.1 

8358.5  1810.0  2645.7  6109.0 

40 

42 

8292.1  1774.9  2571.3  6006.6 

8360.8  1811.2  2648.3  6112.5 

42 

44 

8294.4  1776.1  2573.9  6010.1 

8363.1  1812.4  2650.9  6116  1 

44 

46 

8296.7  1777.3  2576.4  6013.6 

8365.4  1813.6  2653.5  6119.7 

46 

48 

8299.0  1778.5  2578.9  6017.1 

8367.7  1814.9  2656.1  6123.2 

48 

50 

8301.3  1779.7  2581.5  6020.6 

8369.9  1816.1  2658.7  6106  8 

50 

52 

8303.6  1780.9  2584.0  C024.1 

8372.2  1817.3  2661.3  6130.4 

52 

51 

8805.9  1782.1  2586.6  6027.6 

8374.5  1818.5  2663.9  6133.9 

54 

56 

8308.2  1783.3  2589.1  6031.1 

8376.8  1819.7  2666.6  6137.5 

56 

58 

8310.5  1784.5  2591.7  6034.6 

8379.1  1820.9  2669.2  6141.1 

58 

60 

8312.8  1785.7  2594.2  6038.2 

8381.3  1822.2  2671.8  6144.7 

60 

234       FUNCTIONS  OF  A  ONE-DEGREE   CURVE 


94° 

95° 

L.  C.   M.    E.    T. 

L.  C.   M.    E.     T. 

0 

8381.3  1822.2  2671.8  6144.7 

8449.2  1858.9  2751.5  6253.2 

0 

2 

8383.6  1823.4  2674.4  6148.3 

8451.5  1860.1  2754.2  6256.9 

2 

4 

8385.9  1824.6  2677.0  6151.9 

8453.7  1861.3  2756.9  6260.5 

4 

6 

8388.1  1825.8  2679.7  6155.4 

8456.0  1862.6  2759.6  6264.2 

6 

8 

8390.4  1827.0  2G82.3  6159.0 

8458.2  1863.8  2762.3  6267.8 

8 

10 

8392.7  1828.3  2684.9  6162.6 

8460.4  1865.0  2765.0  6271.5 

10 

12 

8395.0  1829.5  2687.6  6166.2 

8462.7  1866.3  2767.7  6275.2 

12 

14 

8397.2  1830.7  2690.2  6169.8 

8464.9  1867.5  2770.4  6278.8 

14 

16 

8399.5  1831.3  2692.8  0173.4 

8467.2  1868.7  2773.1  6282.5 

16 

18 

8401.7  1833.1  2695.6  6177.0 

8469.4  1869.9  2775.8  6286.2 

18 

20 

8404.  C  1834.4  2698.1  6180.6 

8471.7  1871.2  2778.5  6289.8 

20 

22 

8406.3  1835.6  2700.8  6184.2 

8473.9  1872.4  2781.2  6293.5 

22 

24 

8408.5  1836.8  2703.4  6187.8 

8476.2  1873.6  2784.0  6297.2 

24 

26 

8410.8  1838.0  2706.1  6191.5 

8478.4  1874.9  2786.7  6300.9 

26 

28 

8413.1  1839.3  2708.7  6195.1 

8480.7  1876.1  2789.4  6304.6 

28 

30 

8415.3  1840.5  2711.4  6198.7 

8482.9  1877.3  2792.1  6308.2 

30 

32 

8417.6  1841.7  2714.0  6202.3 

8485.1  1878.6  2794.9  6311.9 

32 

31 

8419.9  1842.9  2716.7  6205.9 

8487.4  1879.8  2797.6  6315.6 

34 

36 

8422.1  1844.2  2719.3  6209.5 

8489.6  1881.0  2800.3  6319.3 

36 

38 

8424.4  1845.4  2722.0  6213.2 

8491.9  1882.3  2803.1  6323.0 

38 

40 

8426.6  1846.6  2724.7  6216.8 

8494.1  1883.5  2805.8  6326.7 

40 

42 

8428.9  1847.8  5J727.3  6220.4 

8496.3  1884.8  2808.6  6330.4 

42 

44 

8431.2  1849.1  2730.0  6224.1 

8498.6  1886.0  2811.3  6334.1 

44 

46 

8433.4  1850.3  2732.7  6227.7 

8500.8  1887.2  2814.1  6337.8 

46 

48 

8435.7  1851.5  2735.4  6231.3 

8503.0  1888.5  2816.8  6341.5 

48 

50 

8437.9  1852.7  2738.0  6235.0 

8505.3  1889.7  2819.6  6345.2 

50 

52 

8440.8  1854.0  2740.7  6238.6 

8507.5  1890.9  2822.3  6349.0 

52 

54 

8442.4  1855.2  2743.4  6242.3 

8509.8  1892.2  2825.1  6352.7 

54 

56 

8444.7  1856.4  2746.1  6245.9 

85120  1893.4  2827.8  6356.4 

56 

58 

8447.0  1857.6  2748.8  6249.6 

8514.2  1894.6  2830.6  6360.1 

58 

60 

8449.2  1858.9  2751.5  6253.2 

8516.4  1895.9  2833.4  6363.8 

60 

96° 

97° 

j  / 

L.  C.    M.    E.     T. 

L.  C.    M.    E.     T. 

0 

8516  4  1895.9  2833.4  6363.8 

8583.0  1933.2  2917.5  6476.6 

0 

2 

8518.7  1897.1  2836.1  6367.5 

8585.2  1934.4  2920.3  6480.4 

2 

4 

85209  1898.4  2838.9  6371.3 

8587.5  1935.7  2923.2  6484.2 

4 

6 

8523.1  1899.6  2841.7  6375.0 

8589.7  1936.9  2926.0  6488.0 

6 

8 

8525  4  1900.  S  2844.5  6378.7 

8591.9  1938.2  2928.9  6491.8 

8 

10 

8527.6  1902.1  2847.2  6382.5 

8594  1  1939.4  2931.7  6495.6 

10 

12 

8529  8  1903.3  2850.0  6386.2 

8596.3  1940.7  2934.6  6499.4 

12 

14 

8532.0  1904.6  2852.8  6389.9 

859S.5  1941.9  2937.5  6503.2 

14 

16 

8534.3  1905.8  2855.6  6393.7 

8000.7  1943.2  2940.3  6507.1 

16 

18 

8536.5  1907.0  2858.4  6397.4 

8602.9  1944.4  2943.2  6510.9 

18 

20 

8538.7  1908.3  2861.2  6401.2 

8605.1  1945.7  2946.1  6514.7 

20 

22 

8540  9  1909.5  2864.0  6404.9 

8607.3  1946.9  2948.9  6518.5 

22 

24 

8543.2  1910.8  2866.7  6408.7 

8609.5  1948.2  2951.8  6522.3 

24 

26 

8545.4  1912.0  2869.5  6412.4 

8611.7  1949.4  2954.7  6526.2 

26 

28 

8547  6  1913  3  2872.3  6416.2 

8613.9  1950.7  2957.6  6530.0 

28 

30 

8549.8  1914.5  2875.1  6419.9 

8616.1  1952.0  2960.4  6533.8 

30 

32 

8552  0  1915.7  2877.9  6423.7 

8618.3  1953.2  2963.3  6537.7 

32 

34 

8554.3  1917.0  2880.8  6427  5 

8620.5  1954.5  2966.2  6541.5 

34 

36 

8556.5  1918.2  2883.6  6431.2 

8622.7  1955.7  2969.1  6545.3 

36 

38 

8558.7  1919.5  2886.4  64350 

8624.9  1957.0  2972.0  6549.2 

38 

40 

8560.9  1920.7  2889.2  6438.8 

8627.1  1958.2  2974.9  6553.0 

40 

42 

8563.1  1922.0  2892.0  6442  5 

8629.3  1959  5  2977.8  6556.9 

42 

44 

8565.3  1923.2  2894.8  6446.3 

8631.5  1960.7  2980.7  6560.7 

44 

46 

8567  6  1924  5  2897.7  6450.1 

8633.7  1962.0  2983.6  6564.6 

46 

48 

8569.8  1925.7  2900.5  6453.9 

8635.8  1963.2  2986.5  6568.4 

48 

50 

8572.0  1927.0  2903.3  6457.6 

8638.0  1964.5  2989.4  6572.3 

50 

52 

8574.2  1928.2  2906.!  6461.4 

8040.2  1965.8  2992.3  6576.2 

52 

54 

8576.4  1929  4  2909.0  6465.2 

8642.4  1967.0  2995.2  6580.0 

54 

56 

8578.6  1930.7  2911.8  6469.0 

8044.6  1968.3  2998.1  6583.9 

56 

58 

8580.8  1931.9  2914.7  6472.8 

8646  $  1969.5  3001.1  6587.7 

58 

60 

8583.0  1933.2  2917.5  6476.6 

86490  1970.8  3004  0  6591.6 

60 

FUNCTIONS  OF  A  ONE-DEGREE   CURVE       235 


98° 

99° 

L.  C.    M.     ».    T. 

L.  C.    M.    E.    T. 

0 

8649.0  1970.8  3004.0  6591.6 

8714.3  2008.7  3092.9  6709.0 

0 

2 

8651.2  1972.0  3006.9  6595.5 

8716.4  2009.9  3095.9  6712.9 

2 

4 

8653.3  1973.3  3009.8  6599.4 

8718.6  2011.2  3098.9  6716.9 

4 

6 

8655.5  1974.6  3012.8  6603.2 

8720.7  2012.5  3101.9  6720.8 

6 

8 

8657.7  1975.8  3015.7  6607.1 

8722.9  2013.7  3104.9  6724.8 

8 

10 

8659.9  1977.1  3018.6  6611.0 

8725.1  2015.0  3107.9  6728.8 

10 

12 

8662.1  1978.3  3021.6  6614.9 

8727.2  2016.3  3111.0  6732.7 

12 

14 

8664.3  1979.6  3024.5  6618.8 

8729.4  2017.5  3114.0  6736.7 

14 

16 

8666.4  1980.9  3027.5  6(122.7 

8731.5  2018.8  3117.0  6740.7 

16 

18 

8668.6  1982.1  3030.4  6C26.6 

8733.7  2020.1  3120.0  6744.6 

18 

20 

8670.8  1983.4  3033.3  6630.5 

8735.9  2021.4  3123.1  6748.6 

20 

22 

8673.0  1984.6  3036.3  6634.4 

8738.0  2022.6  3126.1  6752.6 

22 

24 

8675.2  1985.9  3039.3  6638.3 

87<!0.2  2023.9  3129.1  6756.6 

24 

26 

8677.3  1987.2  3042.2  6642.2 

8742.3  2025.2  3132.2  6760.6 

26 

28 

8679.5  1988.4  3045.2  6646.1 

8744.5  2026.4  3135.2  0764.6 

28 

30 

8681.7  1989.7  3048.1  6650.0 

8746.6  2027.7  3138.3  6768.6 

30 

32 

8683.9  1991.0  3051.1  6653.9 

8748.8  2029.0  3141.3  6772.6 

33 

34 

8686.0  1992.2  3054.1  6657.8 

8750.9  2030.3  3144.4  6776.6 

34 

36 

8688.2  1993.5  3057.0  6661.7 

8753.1  2031.5  3147.4  6780.6 

36 

38 

8690.4  1994.7  3060.0  6665.7 

8755.3  2032.8  3150.5  6784.6 

38 

40 

8692.6  1996.0  3063.0  6669.6 

8757.4  2034.1  3153.5  6788.6 

40 

42 

8694.7  1997.3  3066.0  6673.5 

8759.5  2035.4  3156.6  6792.6 

42 

44 

8696.9  1998.5  3068.9  6677.4 

8761.7  2036.6  3159.7  6796.6 

44 

46 

8699.1  1999.8  3071.9  66H1.4 

8763.8  2037.9  3162.7  6800.6 

46 

48 

8701.2  2001.1  3074.9  6685.3 

8766.0  2039.2  3165.8  6804.6 

48 

50 

8703.4  2002.3  3077.9  6689.2 

8768.1  2040.5  3168.9  6808.6 

50 

52 

8705.6  2003.6  3080.9  6693.2 

8770.3  2041.7  3172.0  6812.6 

52 

54 

8707.8  2004.9  3083.9  6697.1 

8772.4  2043.0  3175.1  6816.7 

54 

56 

8709.9  2006.1  3086.9  6701.1 

8774.6  2044.3  3178.1  6820  7 

56 

58 

8712.1  2007.4  3089.9  6705.2 

8776.7  2045.6  3181.2  6824.7 

58 

60 

8714.3  2008.7  3092.9  6709.0 

8778.9  2046.8  3184.3  6828.8 

60 

100° 

101° 

/ 

L  C.    M.    E.    T. 

L.  C.    M.     E.    T. 

/ 

0 

8778.9  2046.8  3184.3  6828.8 

8842.8  2085.3  3278.3  6951.0 

0 

8781.0  2048.1  3187.4  6832.8 

8844.9  2086.6  3281.5  69552 

2 

4 

8783.1  2049.4  3190.5  6836.8 

8847.0  2087.8  3284.7  6959.3 

4 

6 

8785.3  2050.7  3193.6  6840.9 

8849.2  2089.1  3287.9  6963.4 

6 

8 

8787.4  2051.9  3196.7  6844.9 

8851.3  2090.4  3291.1  6967.6 

8 

10 

8789.6  2053.2  3199.8  6849.0 

8853.4  2091.7  3294.3  6971.7 

10 

12 

8791.7  2054.5  3202.9  6853.0 

8855.5  2093.0  3297.5  6975  8 

12 

14 

8793.9  2055.8  3506.0  6857.1 

8857.6  2094.3  3300.7  6980.0 

14 

16 

8796.0  2057.1  3209.1  6861.1 

8859.8  2095.6  3303.9  6984  1 

16 

18 

8798.9  2058.3  3212.2  6865.2 

8861.9  2096.9  3307.1  6988.2 

18 

20 

8800.3  2059.6  3215.4  6869.2 

8864.0  2098.2  3310.3  6992.4 

20 

22 

8802.4  2060.9  3218.5  6873.3 

8866.1  2099.4  3313.5  6996.6 

22 

24 

8804.5  2062.2  3321.6  6877.4 

8868.2  2100.7  3316  7  7000.7 

24 

26 

8806.7  2063.5  3224.7  6881.4 

8870.3  2102.0  3319.9  7004.9 

26 

28 

8808.8  2064.7  3227.9  6885.5 

8872.4  2103.3  3323,.  1  7009.0 

28 

30 

8810.9-  2066.0  3231  0  6889.6 

8874.5  2104.6  3326.4  7013.2 

30 

32 

8813.1  2067.3  3234.1  6893.7 

8876.7  2105.9  3329.6  7017.3 

32 

34 

8815.2  2068.6  3237.3  6897.8 

8878.8  2107.2  3332.8  7021  5 

34 

36 

8817.3  2069.9  3240.4  6901.8 

8880.9  2108.5  3336.0  7025.7 

36 

38 

8819.5  2071.1  3243.5  6905.9 

8883.0  2109.8  3339.3  7029.9 

38 

40 

8821.6  2072.4  3246.7  6910.0 

8885.1  2111.1  3342.5  7034.0 

40 

42 

8823.7  2073.7  3249.8  6914.1 

8887.2  2112.4  3345.8  7038.2 

42 

44 

8825.8  2075.0  3253  C  6918.2 

8889.3  2113.6  3349.0  7042.4 

44 

46 

8828.0  2076.3  3256.2  6922.3 

8891.4  2114.9  3352.3  7046.6 

46 

48 

8830.1  2077.6  3259.3  6926.4 

8893.5  2116.2  3355.5  7050.8 

48 

50 

8832.2  2078.9  3262.5  6930.5 

8895.6  2117.5  3358.8  7055.0 

50 

52 

8834.3  2080.1  3265.7  6934.6 

8897.7  2118.8  3362.0  7059.2 

52 

54 

8836.4  2081.4  3268.8  6938.7 

8899.8  2120.1  3365  5  7063  4 

54 

56 

8838  6  2082.7  ,3272.0  6942.8 

8901.9  2121.4.  3368.7  7067.6 

56 

58 

8340.7  2084.0  3275.2  6946.9 

8904.0  2122.7  3372.0  7071.8 

58 

60 

8842.8  2085.3  3278.3  6951.0 

8906.1  2124.0  3375.1  7076.0 

60 

236       FUNCTIONS  OF  A  ONE-DEGREE  CURVE 


102° 

103° 

/ 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

/ 

0 

8906.1  2124.0  3375.1  7076.0 

8968.7  2163.0  3474.6  7203.6 

0 

2 

8908.2  2125.3  3378.3  7080.2 

8970.8  2164.3  3478.0  7207.9 

2 

4 

8910.3  2126.6  .3381.6  7084.4 

8972.9  2165.6  3481.4  7212.2 

4 

6 

8912.4  2127.9  3384.9  7088.6 

8974.9  2166.9  3484.7  7216  5 

6 

8 

8914.5  2129.2  3388.2  7092.8 

8977.0  2168.2  3488.1  7220.8 

8 

10 

8916.6  2130.5  3391.5  7097.1 

8979.1  2169'.5  3491.5  7225.1 

10 

12 

8918.7  2131.8  33947  7101.3 

8981.1  2170.8  3494.9  7229.5 

12 

14 

8920.8  2133.1  :«98  0  7105.5 

8983.2  2172.1  3498.3  7233.8 

14 

16 

8922.9  2134.4  3101.3  7109.7 

8985.3  2173.4  3501.6  7238.1 

16 

18 

8925.0  2135.7  3104.6  7114.0 

8987.3  2174.7  3505.3  72424 

18 

20 

8927.0  2137.0  3407.9  7118.2 

89894  2176.1  3508.4  7246.8 

20 

22 

8929.1  2138.3  311  1.2  7122.4 

8991.5  2177.4.  3511.8  7251.1 

2*.' 

24 

8931.2  2139.6  3H4.5  7126.7 

8993.5  2178.7  3515.2  7255.4 

24 

26 

8933.3  2140.9  3417.9  7130.9 

8995.6  2180.0  3518.7  7259.8 

26 

28 

893V  4  2142.2  3421.2  7135.2 

8997.7  2181.3  3522.1  7264.1 

28 

30 

8937.5  2143.5  3424.5  7139.4 

8999.7  2182.6  3525.5  7268.5 

30 

32 

8939.6  2144  8  3427.8  7143.7 

9001.8  2183.9  3528.9  7272.8 

32 

34 

8941.6  2146.1  3431.1  7148.0 

9003.9  2185.2  3532.3  7277.2 

34 

36 

8943.7  2147.4  3434.5  7152.2 

9005.9  2186.5  3535.7  7281.5 

36 

38 

8945.8  2148.7  3437.8  7156.5 

9008.0  2187.8  3539.2  7285.9 

38 

40 

8947.9  21500  3441.1  7160.7 

9010.0  2189.1  3542.6  7290.3 

40 

42 

8950.0  2151.3  3444.4  7165.0 

9012.1  2190.5  3546.0  7294.6 

42 

44 

8952.1  2152.6  3447.8  7169.3 

9014.2  2191.8  3549.5  7299.0 

44 

46 

8954.1  2153.9  3451.1  7173.6 

9016.2  2193.1  3552.9  7303.4 

46 

48 

8956.2  2155.2  3454.5  7177.9 

9018.3  2194.4  3556.3  7307.7 

48 

50 

8958.3  2156  5  3457.8  7182.1 

9020.3  2195  7  3559.8  7312.1 

50 

52 

8960.4  2157.8  3461.2  7186.4 

9022.4  2197.0  3563.2  7316.5 

52 

54 

8962.5  2159.1  3464.5  7190.7 

9024.5  2198.3  3566.7  7320.9 

54 

56 

8964  5  2100.4  3467.9  7195.0 

9026.5  2199.6  3570.2  7325.3 

56 

58 

S966.6  2161.7  3471.2  7199.3 

9028.6  2200.9  3573.6  7329.7 

58 

60 

8968.7  2163.0  3474.6  7203.6 

9030.6  2202.3  3577.1  7334.1 

60 

104° 

105° 

L.  C.   M.    E.    T. 

L.  C.   M.    E.    T. 

0 

9030.6  2202.3  3577.1  7334.1 

9091.8  2241.8  3682.6  7467.5 

0 

2 

9032  7  2203.6  3580.5  7338.5 

9093.9  2243.1  3686.1  7472.0 

2 

4 

9034.7  2204.9  3584.0  7342.9 

9095.9  2244.4  3689.7  7476.5 

4 

6 

9036.8  2206.2  3587.5  7347.3 

9097.9  2245.8  3693.3  7481.0 

6 

8 

9038.8  2207.5  3591.0  7351.7 

9099.9  2247.1  3696.9  7485.5 

8 

10 

9040.9  2208  8  3594.4  7356.1 

9102.0  2248.4  3700.4  7490.0 

10 

12 

9042.9  2210  2  3597.9  7360.5 

9104.0  2249.7  3704.0  7494.5 

12 

14 

90450  2211.5  3601.4  7364.9 

9106.0  2251.1  3707.6  7499.1 

14 

16 

9047.0  2212.8  3604.9  7369.4 

9108.0  2252.4  3711.2  7503.6 

16 

18 

9049.1  2214.1  3608.4  7373.8 

9110.1  2253.7  3714.8  7508.1 

18 

20 

9051.1  2215.4  3611.9  7378.2 

9112.1  2255.0  3718.4  7512.6 

20 

22 

9053.1  2216.7  3615.4  7382.6 

9114.1  2256.4  3722.0  7517.2 

22 

24 

9055.2  2218  0  3618.9  7387.1 

9116.1  2257.7  3725.6  7521.7 

24 

26 

9057.2  22194  3622.4  7391.5 

9118.1  22.-9.0  3729.3  7526.3 

26 

28 

9059.3  2220.7  3625.9  7396.0 

9120.2  2260.3  3732.9  7530.8 

28 

30 

9001.3  2222.0  3629.4  7400.4 

9122.2  2261.7  3736.5  7535.3 

30 

32 

9063.3  2223.3  3633.0  7404.8 

9124.2  2263  0  3740.1  7539  9 

32 

34 

9065.4  2224.6  3036.5  74093 

9126.2  2264.3  3743.7  7544.4 

34 

]  36 

9067.4  2226  0  3640.0  7413  8 

9128  2  2265.7  3747.4  7549.0 

36 

38 

9069.5  2227.3  3643.5  7418  2 

9130.2  2267.0  3751.0  75536 

38 

40 

9071.5  2228.6  3647.1  7422.7 

9132.3  2268.3  3754.6  7558.1 

40 

42 

9073  5  2229.9  3650.6  7427.1 

9134.3  2269.6  3758.3  7562.7 

42 

44 

90756  2231  2  3654.1  7431.6 

9136.3  2271.0  3761.9  7567.3 

44 

46 

9077  6  2232.6  3657.7  7436.1 

9138.3  2272.3  3765.6  7571  8 

46 

48 

9079.6  2233.9  3661.2  74406 

9140.3  2273  6  3769.2  7576.4 

48 

50 

9081.7  2235.2  3664.8  7445.0 

9142.3  2275.0  37729  7581.0 

50 

52 

9083.7  2236.5  3668.3  7449.5 

9144.3  2276.3  3776.5  7585.6 

52 

54 

9085.7  2237.8  3671.9  7454.0 

9146  3  2277.6  3780.2  7590.2 

54 

56 

9087.8  2239.2  3675.4  7458  5 

9148.3  2278.9  3783.9  7594.8 

56 

58 

9089.8  2240.5  3679.0  7463.0 

9150.4  2280.3  3787.5  7599.4 

58 

60 

9091  8  2241.8  3682.6  7467  5 

9152  4  2281.6  3791  2  7W4  0 

60 

FUNCTIONS   OF  A  ONE-DEGREE  CURVE       237 


106° 

107° 

/ 

L.  C.   M.    E.    T. 

L.  C.   M.    E.     T. 

0 

9152.4  2-281.6  3791.2  7604.0 

9212.2  2321.7  3903.1  7743.7 

0 

2 

9154.4  228-2.9  3794.9  7608.6 

9214.2  2323.0  3906.9  7748.4 

0 

4 

9156.4  2284.3  3798.6  7613.2 

9216.2  2324.4  3910.7  7753.1 

4 

6 

9158.4  2285.6  3802.3  7617.8 

9218.   2325.7  3914.5  7757.8 

6 

8 

9160.4  2286.9  3805.9  7622.4 

9220.   2327.0  3918.3  7762.5 

8 

10 

9162.4  2288.3  3809.6  7627.0 

9222.   2328.4  3922.1  7767.3 

10 

12 

9164.4  2289.6  3813,3  7631.7 

9224.   2329.7  3925.9  7772.0 

12 

14 

9166.4  2290.9  3817.0  7636.3 

9226.   2331.1  3929.7  7776.7 

14 

l.« 

9168.4  2292.3  3820.7  7610.9 

9228.   233'2.4  3933.6  7781.5 

16 

18 

9170.4  2293.6  3824.4  7645.5 

9230.0  2333.7  3937.4  7786.2 

18 

20 

9172.4  2294.9  3828.1  7650.2 

9232.0  2335.1  3941.2  7791.0 

20 

22 

9174.4  2296.3  3831.9  7654.8 

9234.0  2336.4  3945.0  7795.7 

22 

24 

9176.4  2297.6  3835.6  7659.5 

9235.9  2337.8  3948.9  7800.5 

24 

26 

9178.4  2298.9  3839.3  7664.1 

9-237.9  2339.1  3952.7  7805.2 

26 

28 

9180.4  2300.3  3843.0  7668.8 

9239.9  2340.5  3956.5  7810.0 

28 

30 

9182.4  2301.6  3846.7  7673.4 

9241.9  2341.8  3960.4  7814.7 

80 

32 

9184.4  2302.9  3850.5  7678.1 

9243.8  2343.1  39642  7819.5 

32 

34 

9186.4  2304.3  3854.2  768'.>.7 

9'245.8  2344.5  3968.1  7824.3 

34 

36 

9188.4  2305.6  3858.0  7687.4 

9247.8  2345.8  3971.9  7829.1 

86 

38 

9190.4  2306.9  3861.7  7692.1 

9249.7  2347.2  3975.8  7833.8 

38 

40 

9192.4  2308.3  3865.4  7696.7 

9251.7  2348.5  3979.6  7838.6 

40 

42 

9194.4  2309.6  3869.2  7701.4 

9-253.7  2349.9  3983.5  7843.4 

42 

44 

9196.3  2311.0  3873.0  7706.1 

9255.6  2351.2  3987.4  7848.2 

44 

46 

9198.3  2312.3  3876.7  7710.8 

9257.6  2352.6  3991.3  7853.0 

46 

48 

9200.3  2313.6  3880.5  7715.5 

9259.6  2353.9  3995.1  7857.8 

48 

50 

9202.3  2315.0  3884.2  7720.1 

9261.5  2355.3  3999.0  7862.6 

50 

52 

9204.3  2316.3  3888.0  7724.8 

9263.5  2356.6  4002.9  7867.4 

52 

54 

9206.3  2317.7  3891.8  7729.5 

9265.4  2358.0  4006.8  7872.2 

54 

56 

9208.2  2319.0  3895.6  7734.2 

9267.4  2359.3  4010  7  7877.0 

56 

58 

9210.2  2320.3  3899.3  7739.0 

9269.4  2360.7  4014.6  7881.9 

58 

60 

9212.2  2321.7  3903.1  7743.7 

9-271.8  2362.0  4018.5  7886.7 

60 

108° 

109° 

/ 

L.  C.   M.    E.     T. 

L.  C.   M.    E.     T. 

0 

9271.3  2362.0  4018.5  7886.7 

9329.8  2402.6  4137.4  8033.2 

0 

g 

9273.3  2363.3  4022.4  7891.5 

9331.7  2403.9  4141.4  8038.1 

2 

4 

9275.3  2364.7  40:26.3  7896.3 

9333.6  2405.3  4145.4  8043.1 

4 

6 

9277.2  2366.0  4030.2  7901.2 

9335.6  2406.6  4149.5  8048.0 

6 

8 

9279.2  2367.4  4034.1  7906.0 

9337.5  2408.0  4153.5  8053.0 

8 

10 

9281.1  2368.7  4038.0  7910.8 

9339.4  240S.4  4157.5  8057.9 

10 

12 

9283.1  2370.1  4042.0  7915.7 

9341.4  2410.7  4161.6  8062.9 

12 

14 

9285.0  2371.4  4045.9  7920.5 

9343.3  2412.1  4165.6  8067.9 

14 

16 

9287.0  2372.8  4049.8  7925.4 

9345.2  2413.4  4169.7  8072.8 

16 

18 

9288.9  2374.1  4053.8  7930.3 

9347.2  2414.8  4173.8  8077.8 

18 

20 

9290.9  2375.5  4057.7  7935.1 

9349.1  2416.2  4177.8  8082.8 

20 

22 

9292.8  2376.8  4061.6  7940.0 

9351.0  2417.5  4181.9  8087.8 

22 

24 

9294.8  2378.2  4065.6  7944.8 

9352.9  2418.9  4186.0  8092.8 

24 

26 

9296.7  2379.5  4069.5  7949.7 

9354.9  2420.2  4190.0  8097.8 

26 

28 

9298.7  2380.9  4073.5  7954.6 

9356.8  2421.6  4193.1  >  8102.8 

28 

30 

9300.6  2382.3  4077.5  7959.5 

9358.7  2423.0  4198.2  8107.8 

30 

82 

9302.6  2383.6  4081.4  7964.4 

9360.6  2424.3  4202.3  8112.8 

32 

34 

9304.5  2385.0  4085.4  7969.3 

9362.6  2425.7  4206.4  8117.8 

34 

36 

9306.5  2386.3  4089.4  7974.1 

9364.5  2427.0  4210.5  8122.8 

36 

38 

9308.4  2387.7  4093.4  7979.0 

9366.4  2428.4  4214.6  8127.8 

38 

40 

9310.4  2389.0  4097.3  7983.9 

9368.3  2430.0  4218.7  8132.8 

40 

42 

9312.3  2390.4  4101.3  7988.8 

9370.2  2431.1  4222.8  8137.9 

42 

44 

9314.2  2391.7  4105.3  7993.8 

9372.2  2432.5  4'226.9  8142.9 

44 

46 

9316.2  2393.1  4109.3  7998.7 

9374.1  2433.9  4231.0  8147.9 

46 

48 

9318.1  2394.4  4113.3  8003.6 

9376.0  2435.2  4235.1  8153.0 

48 

50 

9320.1  2395.8  4117.3  8008.5 

9377.9  2436.6  4239.3  8158.0 

50 

52 

9322.0  2397.2  4121.3  8013.4 

9379.8  2438.0  4'243.4  8163.1 

52 

54 

9323.9  2398.5  4125.3  8018.4 

9381.7  2439.3  4247.5  8168.1 

54 

56 

9325.9  2399.9  4129.3  8023  3 

9383.7  2440.7  4251.7  8173.2 

56 

58 

9327.8  2401.2  4133.4  8028.2 

9385.6  2442.1  4255.8  8178.2 

58 

60 

9320.8  240-2.6  4137.4  8033.2 

9387.5  24434  4260.0  8183.3 

60 

238       FUNCTIONS   OF  A  ONE-DEGREE  CURVE 


110° 

111° 

L.  C.   M.     E.    T. 

L.  C.    M.    E.     T. 

0 

9387.5  2443.4  4260.0  8183.3 

9444.5  2484.5  4386.4  8337.2 

0 

2 

9389.4  2444.8  4264.1  8188.4 

9446.4  2485.9  4390.7  8342.4 

2 

4  ' 

9391.3  2446.1  4268.3  8193.4 

9448.3  2487.2  4395  0  8347.6 

4 

6 

9393  2  2447.5  4272.4  8198.5 

9450.1  2488.6  4399.3  8352.8 

6 

8 

9395.1  2448.9  4276.6  8203.6 

9452.0  2490.0  4403.6  8358.0 

8 

10 

9397.0  2450.2  4280.8  8208.7 

9453.9  2491.4  4407.9  8363.2 

10 

12 

9398.9  2451.6  4284.9  8213.8 

9455.8  2492.7  4412.  "2  8368.5 

12 

14 

9400.8  2453.0  4289.1  8218.9 

9457.7  2494.1  4416.5  8373.7 

14 

16 

9402.7  2454.3  4293.3  8224.0 

9459.6  2495.5  4420.8  8378  9 

16 

18 

9404.7  2455.7  4297.5  8229.1 

9461.4  2496.9  4425.1  8384  1 

18 

20 

9406.6  2457.1  4301.7  8234.2 

9463.3  2498.2  4429.5  8389.4 

20 

22 

9408.5  2458.4  4305.9  8239.3 

9465.2  2499.6  4433  8  8394  6 

22 

24 

9410.4  24598  4310.1  8244.4 

9467.1  2501.0  4438.1  83£9.9 

24 

26 

9412.3  2461.2  4314.3  8249.5 

9469.0  2502.4  4442.5  8405.1 

26 

28 

9414.2  2462.6  4318.5  8254.6 

9470.8  2503.8  4446.8  8410.4 

28 

30 

9416.1  2463.9  4322.7  8259.8 

9472.7  2505.1  4451.2  8415.6 

30 

'6-2 

9418.0  2465.3  4326.9  8:264.9 

9474.6  2506.5  4455.5  84^0  9 

32 

34 

9419.9  2466.7  4331.1  8270.0 

9476.5  2507.9  4459.9  8426.2 

34 

36 

9421.8  2468.0  4335.4  8275.2 

9478.3  2509.3  4464.2  8431.4 

36 

38 

9423.7  2469.4  4339.6  8280.3 

9480.2  2510.6  4468.6  8436.7 

38 

40 

9425.6  2470.8  4343.8  8285.5 

9482.1  2512.0  4473.0  8442.0 

40 

42 

9427.5  2472.1  4348.1  8290.6 

9484.0  2513.4  4477.3  8447.3 

42 

44 

9429.3  2473.5  4352.3  8295.8 

9485.8  2514.8  4481.7  8452.6 

44 

46 

9431.2  2474.9  4356.6  8300.9 

9487.7  2516.2  4486.1  8457.9 

46 

48 

9433.1  2476.3  4360.8  8306.1 

9489.6  2517.5  4490.5  8463.2 

48 

50 

9435.0  2477.6  4365.1  8311.3 

9491.4  2518.9  4494.9  8468.5 

50 

52 

9436.9  2479.0  4369.3  8316.5 

9493.3  2520.3  4499.3  8473.8 

52 

54 

9438.8  2480.4  4373.6  8321.6 

9495.2  2521.7  4503.7  8479.1 

54 

56 

9440.7  2481.7  4377.9  8326.8 

9497.0  2523.1  4508.1  8484.4 

56 

58 

9442.6  2483.1  4382.2  8332.0 

9498.9  2524.5  4512.5  8489.7 

58 

60 

9444.5  2484  5  4386,4  8337.2 

9500.8  2525.8  4516.9  8495.1 

60 

112° 

113° 

L.  C.   M.    E.     T. 

L.  C.   M.     E     T. 

0 

9500.8  2525.8  4516.9  8495.1 

9556.3  2567.4  4651.6  8657.1 

0 

2 

9502.6  2527.2  4521.4  8500.4 

9558.2  2568.8  4656.2  8662.6 

2 

4 

9504.5  2528.6  4525.8  8505.8 

9560.0  2570.2  4660.8  8668.0 

4 

6 

9506.4  2530.0  4530.2  8511.1 

9561.8  2571.6  4665.4  8673.5 

6 

8 

9508.2  2531.4  4534.6  8516.4 

9563.7  2573.0  4669.9  8679.0 

8 

10 

9510.1  2532.7  4539.1  8521.8 

9565.5  2574.4  4674.5  8684.5 

10 

12 

9511.9  2534.1  4543.5  8527.1 

9567.4  2575.8  4679.1  8690.0 

12 

14 

9513.8  2535.5  4548.0  8532.5 

9569.2  2577.1  4683.7  8695.5 

14 

16 

9515.7  2536.9  4552.4  8537.9 

9571.0  2578.5  46883  8701.0 

16 

18 

9517.5  2538.3  4556.9  8543.2 

9572.9  2579.9  4692.9  8706.5 

18 

20 

9519.4  2539.7  4561.3  8548.6 

9574.7  2581.3  4697.5  8712.0 

20 

22 

9521.2  2541.0  4565.8  8554.0 

9576.5  2582.7  4702.1  8717.6 

22 

24 

9523.1  2542.4  4570.3  8559.4 

9578.4  2584.1  4706.8  8723.1 

24 

26 

9524.9  2543.8  4574.8  8564.8 

95802  2585.5  4711.4  8728.6 

26 

28 

9526.8  2545.2  4579.3  8570.2 

9582.0  2586.9  4716.0  8734.2 

28 

30 

9528.6  2546.6  4583.7  8575.6 

9583.8  2588.3  4720.6  8739.7 

30 

32 

9530.5  2548.0  4588.2  8581.0 

9585.7  2589.7  4725.3  8745.3 

32 

34 

9532.3  2549.4  4592.7  8586.4 

9587.5  2591.1  4729.9  8750.8 

34 

36 

9534.2  2550.7  4597.2  8591.8 

9589.3  259-2.5  4734.6  8756.4 

36 

38 

9536.0  2552.1  4601.7  8597.2 

9591.1  2593.9  4739.2  8761.9 

38 

40 

9537.9  2553.5  4606  2  8602.6 

9593.0  2595.3  4743.9  8767.5 

40 

42 

9539.7  2554.9  4610.8  8608.0 

9594.8  2596.7  4748.5  8773.1 

42 

44 

9541.6  2556.3  4615.3  8613.5 

9596.6  2598.1  4753.2  8778.6 

44 

46 

9543.4  2557.7  4619.8  8618.9 

9598.4  2599.4  4757.9  8784.2 

46 

48 

9545.3  2559.1  4624.3  8624.3 

9600.3  2600.8  4762.6  8789.8 

48 

50 

9547.1  2560.5  4628.9  8629.8 

9602.1  2602.2  4767.2  8795.4 

50 

52 

9549.0  2561.8  4633.4  8035.2 

9603.9  2603.6  4771.9  8801.0 

52 

54 

9550.8  2563  2  4638.0  8640.7 

9605.7  2605  0  4776.6  8806.6 

54 

56 

9552  6  2564.6  4642.5  8646.2 

9607.5  2606.4  4781.3  8812.2 

56 

58 

<>554.5  2566.0  4647.1  8651.6 

9609.4  2607.8  4786.0  8817.8 

58 

60 

95563  2567.4  4651.6  8657.1 

9611.2  2609.2  4790.7  8823.4 

60 

FUNCTIONS  OF  A  ONE-DEGREE  CURVE       239 


i  . 

114° 

116° 

/ 

L.  C.    M.    E.    T. 

L.  0.    M.    E.    T. 

0 

9611  2  2609.2  4790.7  8823.4 

9665.3  2651.3  4934.4  8994.3 

0 

2 

9613.0  2610.6  4795.5  8829.1 

9667.1  2652.7  4939.3  9000.1 

2 

4 

9614.8  2612.0  4800.2  8834.7 

9668.8  2654.1  4944.2  9005.9 

4 

6 

9616.6  2613.4  4804.9  8840.3 

9670.6  2655.5  4949.1  9011.6 

6 

8 

9618.4  2614.8  4809.6  8846.0 

9672.4  2656.9  4954.0  9017.4 

8 

10 

9620.2  2616.2  4814.4  8851.6 

9674.2  2658.3  4958.9  9023.2 

10 

12 

9622.0  2617.6  4819.1  8857.2 

9676.0  2659.7  4963.8  9029.0 

12 

14 

9623.8  2619.0  4823.9  8862.9 

9677.8  2661.1  4968.7  9034.8 

14 

16 

9625.7  2620.4  4828.6  8868.5 

9679.6  2662.5  4973.6  9040.7 

16 

18 

9627.5  2621.8  4833.4  8874.2 

9681.4  2663.9  4978.5  9046.5 

18 

20 

9629.3  2623.2  4838.1  8879.9 

9683.1  2665.4  4983.4  9052.3 

20 

22 

9631.1  2624.6  4842.9  8885.5 

9684.9  2666.8  4988.3  9058.1 

22 

24 

9632.9  2626.0  4847.7  8891.2 

9686.7  2668.2  4993.3  9064.0 

24 

26 

9634,7  2627.4  4852.4  8896.9 

9688.5  2669.6  4998.2  9069.8 

26 

28 

9636.5  2628.8  4857.2  8902.6 

9690.3  2671.0  5003.2  9075.7 

28 

30 

9638.3  2630.2  4862.0  8908.3 

9692.0  2672.4  5008.1  9081.5 

30 

32 

9640.1  2631.6  4866.8  8914.0 

9693.8  2673.8  5013.1  9087.4 

32 

34 

9641.9  2633.0  4871.6  8919.7 

9695.6  2675.2  5018.0  9093.2 

84 

36 

9643.7  2634.4  4876.4  8925.4 

9697.4  2676.6  5023.0  9099.1 

36 

38 

9645.5  2635.8  4881.2  8931.1 

9699.1  2678.0  5028.0  9105.0 

38 

40 

9647.3  2637.2  4885.0  8936.8 

9700.9  2679.5  5032.9  9110.8 

40 

42 

9649.1  2638.6  4890.9  8942.6 

9702.7  2680.9  5037.9  9116.7 

42 

44 

9650.9  2640.0  4895.7  8948.3 

9704.5  2682.3  5042.9  9122.6 

44 

46 

9652.7  2641.4  4900.5  89540 

9706.2  2683.7  5047  9  9128.5 

46 

48 

9654.5  2642.9  4905.3  8959.8 

9708.0  2685.1  5052.9  9134.4 

48 

50 

9656.3  2644.3  4910.2  8965.5 

9709.8  2686.5  5057.9  9140.3 

50 

52 

9658.1  2645.7  4915.0  8971.3 

9711.6  2687.9  5062.9  9146.2 

5',? 

54 

9659.9  2647.1  4919.9  8977.0 

9713.3  2689.3  5067.9  9152.1 

54 

56 

9061.7  2648.5  4984.7  8982.8 

9715.1  2690.7  5072.9  9158.1 

56 

58 

9663.5  2649.9  4929.6  8988.5 

9716.9  2692.2  5078.0  9164.0 

58 

60 

9665.3  2651.3  4934.4  8994.3 

9718.6  2693.6  5083.0  9169.9 

60 

116° 

117° 

L/C.    M.    E.    T. 

L.  C.    M.    E.    T. 

0 

9718.6  2693.6  5083.0  9169.9 

9771.3  2736.1  5236.6  9350.5 

0 

9720.4  2695.0  5088.0  9175.9 

9773.0  2737.5  5241.8  9356.6 

2 

4 

9722.2  2696.4  5093.1  9181.8 

9774.7  2788.9  5247.0  9362.7 

4 

6 

9723.9  2697.8  5098.1  9188.8 

9776.5  2740.4  5252.2  9368.9 

6 

8 

9725.7  2699.2  5103.2  9193.7 

9778.2  2741.8  5257.4  9375.0 

8 

10 

9727.4  2700.6  5108.2  9199.7 

9779.9  2743.2  5262.6  9381.1 

10 

12 

9729.2  2702.1  5113.3  9205.6 

9781.7  2744.6  5267.9  9387.3 

12 

14 

9731.0  2703.5  5118.4  9211.6 

9783.4  2746.0  5273.1  9393.4 

14 

16 

9732.7  2704.9  5123.4  9217.6 

9785.2  2747.5  5278.4  9399.5- 

16 

18 

9734.5  2706.3  5128.5  9223.6 

9;86.9  2748.9  5283.6  9405.7 

18 

20 

9736.3  2707.7  5133.6  92296 

9788.6  2750.3  5288.9  9411.9 

20 

22 

9738.0  2709.1  5138.7  9235  5 

9790.4  2751.7  5294.2  9418.0 

22 

24 

9739.8  2710.6  5143.8  9241.5 

9792.1  2753.2  5299.5  9424.2 

24 

26 

9741.5  2712.0  5148.9  9247.6 

9793.8  2754.6  5304.7  9430.4 

26 

28 

9743.3  2713.4  5154  0  9">53.6 

9795.6  2756.0  5310.0  9436.6 

28 

30 

9745.0  2714.8  5159.1  9259.6 

9797.3  2757.4  5315:3  9442.8 

30 

32 

9746.8  2716.2  5164.2  9265  6 

9799.0  2758.9  5320.6  9449.0 

32 

34 

9748.5  2717.6  5169.4  9271.6 

9800.7  2760.3  5325.9  9455.2 

34 

36 

9750.3  2719.1  5174.5  9277.7 

9802.5  2761.7  5331.2  9461.4 

36 

38 

9752.0  2720.5  5179.7  9283.7 

9804.2  2763.1  5336.5  9467.6 

38 

40 

9753.8  2721.9  5184.8  9289.8 

9805.9  2764.6  5341.8  9473.8 

40 

42 

9755.6  2723.3  5190.0  9295.8 

9807.7  2766.0  5347.2  9480.0 

42 

44 

9757.3  2724.7  5195.1  9301.9 

9809.4  8767.  4  5352.5  9486.3 

44 

46 

9759.0  2726.2  5200.3  9307.9 

9811.1  2768.8  5357.9  9492.5 

46 

48 

9760.8  2727.6  5205.4  9314.0 

9812.8  2770.3  5363.2  9498.7 

48 

60 

9762.5  2729.0  5210.6  9320.1 

9814.5  2771.7  5368.5  9505.0 

50 

52 

9764.3  2730.4  5215.8  9326.1 

9816.3  2773.1  5373.9  9511.2 

52 

54 

9766.0  2731.8  5221.0  9332.2 

9818.0  2774.6  5379.3  9517.5 

54 

56 

9767.8  2733.3  5226.2  9338.3 

9819.7  2776.0  53S4.7  9523.8 

56 

58 

9769.5  2734.7  5231.4  9344.4 

9821.4  2777.4  5390.0  9530.0 

58 

60 

9771.3  2736.1  5236.6  93505 

9823.1  2778.8  5395.4  9536.3 

60 

240    NATURAL  SINES,   COSINES,   TANGENTS,   COTAN. 
0°  1° 


Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

r 

o 

1 

2 
3 
JL_ 
5 
6 

8 

g 

.00000 
.00029 
.00058 
.00087 
.00116 

One 
One 
One 
One 
One 

•00000 
.00029 
.00058 
.00087 
.00116 

Infinite 
3437-75 
1718-87 
1145-92 
839-436 

•01745 
.01774 
.01803 
.01832 
.01862 
.01891 
-01920 
.01949 
.01978 
.02007 

.99985 

-99984 
.99984 
.99983 
.99983 

.01746 
.01775 
-01804 
.01833 
.01862 

57.2900 
56-3506 
55-4415 
54-5613 
53-7086 

60 

59 
58 

57 
58 

.00145 
.00175 
.00204 
.00233 
.00262 

One 
One 
One 
One 
One 

.00145 
.00175 
•00204 
.00233 
.00262 

687-549 
572-957 
491-106 
429-718 
381.971 

-99982 
-99982 
-99981 
-99980 
^99980 
~99979 
-99979 
.99978 
-99977 
.99977 

•01891 

-01920 
.01949 
.01978 
•02007. 
.02036 
-02066 
-02095 
.02124 
.02153 

52-8821 
52-0807 
51-3032 
50-5485 
49-8157 

55 
54 
53 
52 
51 

10 

11 

12 
13 

!i_ 

15 
16 
17 
18 
19 
20 
21 
22 
23 
24 

.00291 
.00320 
.00349 
.00378 
.00407 

One 
.99999 
.99999 
.99999 
.99999 

.00291 
.00320 
.00349 
.00378 
.00407 

343  .  774 

312-521 
286.478 
264-441 
245-552 

.02036 
.02065 
-02094 
.02123 
.02152 

49-1039 
48-4121 
47-7395 
47-0853 
46-4489 

50 

49 
48 

47 
46 

.00436 

.00465 
.00495 
.00524 
.00553 

.99999 
.99999 
.99999 
.99999 
.99998 

.00436 
.00465 
.00495 
-00524 
.00553 

229.182 
214.858 
202.219 
190-984 

180-932 

•02181 
.02211 
.02240 
-02269 
.02298 

,99976 
.99976 
-99975 
.99974 
.99974 

.02182 

.02211 
.02240 
.02269 
.02298 

45-8294 
45-2261 
44-6386 
44-0661 
43-5081 

45 
44 
43 
42 

-i! 

40 

39 
38 
37 
_36 
35 
34 
33 
32 
31 

.00582 
.00611 
.00640 
.00689 
.00698 

.99998 
.99998 
.99998 
.99998 
.99998 

•00582 
.00611 
.00640 
.00869 
.00698 

171-885 
163-700 
156-259 
149-465 
143-237 

-02327 
.02356 
-02385 
.02414 
.  02443__ 
^02472 
.02501 
-02530 
.02560 
.02589 

.99973 
.99972 
.99972 
.99971 
.99970 

.99969 
.99969 
-99968 
.99967 
.99966 

•02328 
.02357 
.02386 
.02415 
.02444 

42-9641 
42-4335 
41-9158 
41-4106 
40-9174 

25 
26 
27 
28 
2P 

•00727 
.00756 
00785 
.00814 
.00844 

.99997 
.99997 
.99997 
.99997 
.99996 

.00727 
.00756 
.00785 
.00815 
.OC844 

137-507 
132.219 
127-321 
122.774 
118-540 

.02473 

.02502 
.02531 
.02560 
.02589 

40-4358 
39-9655 
39.5059 
39-0568 
38-6177 

30 

31 
32 
33 
34_, 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

.00873  i.  99996 
.00902  -99993 
.00931  .99998 
.00960  .99995 
.00989  .99995 

.00873 
.00902 
.00931 
.00960 
.00989 

114.589 
110-892 
107-426 
104-171 
101-107 

.02618 

-02647 
-02676 
.02705 
.02734 

.99966 
-99965 
-99964 
-99963 
.99963 

.02619 
.02648 
.02677 
.02706 
.02735 

38-1885 
37.7686 
37-3579 
36-9560 
36-5627 
36-1776 
35-8006 
35-4313 
35-0695 
34-7151 

30 

29 
28 
27 
26 

.01018 
.01047 
.01076 
.01105 
.01134 

.99995 
.99995 
.99994 
.99994 
.99994 

.01018 
.01047 
.01076 
.01105 
.01135 

98-2179 
95-4895 
92.9085 
90.4633 
88-1436 

-02763 
-02792 
-02821 
-02850 
.02879 

.99962 
.99961 
.99960 
.99959 
-99959 

.02764 
.02793 
.02822 
.02851 
.02881 

25 
24 
23 
22 
21 

.01164 
.01193 
.01222- 
.01251 
.01280 

.99993 
.99993 
.99993 
.99992 
.99992 

.01164 
.01193 
.01222 
.01251 
.01280 

85-9398 
83-8435 
81-8470 
79-9434 
78.1263 

•02908 
-02938 
.02967 
•02996 
-03025 

•99958 
-99957 
.99956 
.99955 
.99954 

.02910 
.02939 
.02968 
.02997 
.03026 

34-3678 
34-0273 
33-6935 
33-3662 
33-0452 

20 

19 
18 
17 
-IB 
15 
14 
13 
12 

_!! 

10 

9 
8 
7 
6 

45 
46 
47 
48 
49 

.01309 
.01338 
.01367 
.01396 
.01425 

.99991 
.99991 
.99991 
.99990 
.99990 

•01309 
.01338 
.01367 
.01396 
•01425 
.01455 
.01484 
.01513 
.01542 
.01571 

76-3900 
74-7292 
73-1390 
71.6151 
70.1533 

•03054 
-03083 
•03112 
•03141 
.03170 

.99953 
.99952 
-99952 
.99951 
.99950 
.99949 
.99948 
.99947 
.99946 
.99945 
.99944 
.99943 
.99942 
.99941 
.99940 

•03055 
.03084 
-03114 
-03143 
.03172 

32-7303 
32-4213 
32-1181 
31-8205 
31-5234 

50 

51 
52 
53 
54 

.01454 
.01483 
•01513 
•01542 
.01571 

.99939 
.99989 
.99989 
.99988 
.99988 

68-7501 
67-4019 
66.1055 
64.8580 
63.6567 

•03199 
.03228 
.03257 
.03236 
.03316 

.03201 
.03230 
-03259 
-03288 
03317 

31-2416 
30-9599 
30.6833 
30-4116 
30.1446 

55 
56 
57 
58 
59 

00 

.01600 
•01629 
•  01658 
•  01687 
•01716 

.99987 
.99987 
.99988 
.99986 
.99985 

.01600 
.01629 
.01658 
.01687 
.01716 

62.4992 
61-3829 
60.3058 
59.2659 
58-2612 

•03345 

.03374 
-03403 
-03432 
.03461 

-03346 
03376 
03405 
03434 
.03463 

29.8823 
29-6245 
,?9.3711 
29.1220 
28-8771 

5 
4 
3 
2 
1 

•01745 

.99985 

.01746 

57-2900 

•03490 

.99939 

03492 

28-6363 

O 

Cos. 

Sin. 

Cot.     Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

89° 


88° 


NATURAL  SINES,   COSINES,   TANGENTS,   COTAN.     241 


2° 


3° 


/ 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

/ 

0 

2 
3 
4 

.03490 
.03519 
.03548 
.03577 
.03606 

-99939 
•99938 
.99937 
.99936 
.99935 

-03492 
.03521 
.03550 
.03579 
.03609 

28-6363 
28.3994 
28-1664 
27-9372 
27-7117 

-05234 
.05263 
-05292 
-05321 
.05350 

.99863 
.99861 
.99860 
.99858 
•99857 

.05241 
.05270 
.05299 
.05328 
.05357 

19-0811 
18-9755 
18-8711 
18-7678 
18.6656 

6O 

59 
58 
57 
56 

5 
6 
7 
8 
9 

.03635 
.03664 
.03693 
.03723 
.03752 

.99934 
.99933 
.99932 
.99931 
.99930 

.03638 
.03667 
.03696 
.03725 
.03754 

27-4899 
27.2715 
27-0566 
26.8450 
26-6367 

.05379 
.05408 
.05437 
-05466 
.05495 

.99855 
.99854 
.99852 
.99851 
.99849 

.05387 
.05416 
.05445 
.05474 
-05503 

18-5645 
18-4645 
18-3655 
18-2677 
18-1708 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.03781 
.03810 
.03839 
.03868 
.03897 

.99929 
.99927 
.99926 
.99925 
.99924 

.03783 
.03812 
.03842 
.03871 
.03900 

26-4316 
26-2296 
26-0307 
25.8348 
25-6418 

-05524 
.05553 
.05582 
.05611 
.05640 

.99847 
.99846 
.99844 
.99842 
.99841 

.05533 
.05562 
.05591 
.05620 
.05649 

18.0750 
17-9802 
17-8863 
17-7934 
17-7015 

5O 

49 
48 
47 
46 

15 
16 
17 
18 
19 

.03926 
.03955 
.03984 
.04013 

.04042 

.99923 
.99922 
.99921 
.99919 
.99918 

.03929 
-03958 
.03987 
.04016 
.04046 

25-4517 
25-2644 
25-0798 
24-8978 
24-7185 

.05669 
.05698 
-05727 
-05756 
.05785 

.99839 
.99838 
.99836 
.99834 
-99833 

.05678 
.05708 
.05737 
.05766 
.05795 

17-6106 

17-5205 
17-4314 
17-3432 

17-2558 

45 
44 
43 
42 
41 

20 

21 
22 
23 

?A 

.04071 
.04100 
.04129 
.04159 
.04188 

-99917 
-99916 
-99915 
.99913 
.99912 

-04075 
-04104 
-04133 
-04162 
-04191 

24-5418 
24-3675 
24-1957 
24-0263 
23.8593 

-05814 
.05844 
-05873 
-05902 
.05931. 

-99831 
-99829 
-99827 
-99826 
-99824 

.05824 
-05854 
-05883 
.05912 
-05941 

17-1693 
17-0837 
16-9990 
16-9150 
16-8319 

4O 

39 
38 
37 
36 

25 
26 
27 
28 
29 

.04217 
.04246 
.04275 
.04304 
.04333 

.99911 
.99910 
-99909 
.99907 
.99906 

.04220 
.04250 
.04279 
-04308 
-04337 

23-6945 
23-5321 
23-3718 
23-2137 
23-0577 

-05960 
-05989 
-06018 
.06047 
.06076 

-99822 
.99821 
.99819 
.99817 
.99815 

.05970 
.05999 
.06029 
.06058 
.06087 

16.7496 
16-6681 
16-5874 
16-5075 
16-4283 

35 

34 
33 
32 
31 

30 

31 
32 
33 
84 

•04362 
-04391 
.04420 
-  04449 
.04478 

-99905 
-99904 
-99902 
-99901 
-99900 

.04366 
.04395 
.04424 
.04454 
-04483 

22-9038 
22-7519 
22-6020 
22-4541 
22-3081 

.06105 
.06134 
.06163 
.06192 
-06221 

.99813 
-99812 
-99810 
-99808 
-99806 

.06116 
-06145 
-06175 
.06204 
-06233 

16-3499 
16-2722 
16.1952 
16.1190 
16-0435 

30 

29 
28 
27 
26 

85 

36 
37 
38 
39 

.04507 
.04536 
.04565 
-04594 
.04623 

-99898 
.99897 
.99896 
-99894 
-99893 

-04512 
-04541 
-04570 
-04599 

.04628 

22J  640 
22-0217 
21-8813 
21-7426 
21-6056 

-06250 
.06279 
-06308 
.06337 
.06366 

-99804 
-99803 
.99801 
.99799 
.99797 

.06262 
-06291 
.06321 
-06350 
-06379 

15-9687 
15-8945 
15-8211 
15-7483 
15-6762 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

•04653 
.04682 
.04711 
.  04740 
.04769 

-99892 
-99890 
-99889 
•99888 
.99886 

.04658 
.04687 
-04716 
-  04745 
-04774 

21-4704 
21-3369 
21-2049 
21-0747 
20-9460 

-06395 
•06424 
•06453 
•06482 
-06511 

.99795 
-99793 
-99792 
-99790 
-99788 

-06408 
-06437 
-06467 
•06496 
-06525 

15-6048 
15-5340 
15-4638 
15-3943 
15-3254 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

-04798 
-04827 
.04856 
-04885 
-04914 

-99885 
.99883 
.99882 
-99881 
99879 

-04803 
.04833 
04862 
-04891 
.04920 

20-8188 
20-6932 
20-5691 
20-4465 
20-3253 

-06540 
.06569 
•06598 
.06627 
.06656 

-99786 
-99784 
-99782 
-99780 
-9.9778 

-06554 
.06584 
-06613 
-06642 
.06671 

15-2571 
15-1893 
15-1222 
15-0557 
14-9898 

15 

14 
13 
12 
11 

50 

51 
52 
53 

54 

-04943 
-04972 
-05001 
-05030 
.05059 

-99878 
-99876 
-99875 
-99873 
-99872 

.04949 
-04978 
-05007 
-05037 
.05066 

20-2056 
20-0872 
19-9702 
19-8546 
19-7403 

-06685 
-06714 
-06743 
-06773 
-06802. 

.99776 
-99774 
.99772 
.99770 
.99768 

-06700 
-06730 
.06759 
-06788 
.06817 

14.9244 
14-8596 
14-7954 
14.7317 
14-6685 

10 

9 

7 
3 

55 
56 

U 

99 

-05088 
-05117 
-05146 
-05175 
-05205 

-99870 
-99869 
.99867 
-99886 
.99864 

-05095 
-05124 
-05153 
-05182 
-05212 

19-6273 
19-5156 
19-4051 
19-2959 
19-1879 

-06831 
.06860 
.06889 
.06918 
-06947 

.99766 
-99764 
-99762 
.99760 
.99758 

.06847 
.06876 
.06905 
.06934 
-06963 

14-6059 
14-5438 
14.4823 
14.4212 
14.3607 

5 
3 

1 

60 

.05234 

,99863 

-05241 

19-0811 

.06976 

-99756 

-OR993 

14-3007 

0 

-7~ 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

* 

87° 


86° 


242    NATURAL  SINES,   COSINES,   TANGENTS,   COTAN. 


4° 


5° 


t 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tari. 

Cot. 

/ 

? 

2 

JL 

.06976 
.07005 

.07034 
.07063 
.07092 

•99756 
•99754 

.99752 
.99750 
.99748 

.06993 
•07022 
.07051 
.07080 
.07110 

14-3007 
14.2411 
14.1821 
14-1235 
14.0655 

•08716 
.08745 

.08774 
.08803 
-08831 

.99619 
.99617 
.99614 
-99612 
.99609 

-08749 
-08778 
-08807 
.08837 
.08866 

11.4301 
11.3919 
11.3540 
11.3163 
11.2789 

60 

59 
58 
57 
56 

6 
J 

J 

.07121 
.07150 
.07179 
.07208 

.07237 

.99746 
.99744 
.99742 
.99740 
.99738 

.07139 
.07168 
.07197 
.07227 
.07256 

14-0079 
13-9507 
13.8940 
13-8378 
13-7821 

•08860 
.08889 
.08918 
.08947 
.08976 

.99607 
.99604 
.99602 
.99599 
.99596 

•08895 
•08925 
-08954 
-08983 
-09013 

11-2417 
11-2048 
11-1681 
11-1316 
11.0954 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.07266 
.07295 
.07324 
.07353 
.07382 

.99736 
.99734 
.99731 
.99729 
.99727 

.07285 
.07314 
.07344 
.07373 
.07402 

13-7267 
13.6719 
13-6174 
13-5634 
13-5098 

-09005 
.09034 
.09063 
.09092 
.09121 

.-99594 
•99591 
.99588 
.99586 
.99583 

•09042 
•09071 
-09101 
.09130 
•09159 

11.0594 
11-0237 
10-9882 
10-9529 
10-9178 

50 

49 
48 
47 
4ft 

15 
16 
17 
18 
19 

.07411 
.07440 
-07469 
.07498 
.07527 

.99725 
.99723 
.99721 
.99719 
.99716 

.07431 
.07461 
.07480 
.07519 
.07548 

13  4566 
13-4039 
13.3515 
13-2996 
13-2480 

.09150 
-09179 
.09208 

.09237 
.09266 

-99580 
.99578 
.99575 
.99572 
.99570 

.09189 
.09218 
.09247 
.09277 
.09306 

10-8829 
10-8483 
10-8139 
10-7797 

10.7457 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

.07556 
.07585 
.07614 
.07643 
.07672 

.99714 
.99712 
.99710 
.99708 

.99705 

.07578 

.07607 
.07636 
.07665 
.07695 

13-1969 
13.1461 
13-0958 
13-0458 
12-9962 

.09295 
.09324 
-09353 
.09382 
.09411 

.99567 

.99564 
.99562 
.99559 
-99556 

-09335 

-09365 
.09394 
.09423 
.09453 

10.7119 
10.6783 
10-6450 
10-6118 
10.5789 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

.07701 
.07730 
.07759 
.07788 
.07817 

.99703 
.99701 
.99699 
.99696 
.99694 

.07724 
.07753 
.07782 
.07812 
.07841 

12-9469 
12-8981 
12-8496 
12-8014 
12.7536 

.  09440 
.09469 
.09493 
.09527 
-09556 

.99553 
.99551 
.99548 
.99545 
.99542 

.09482 
.09511 
.09541 
.09570 
.09600 

10  5462 
10-5136 
10.4813 
10-4491 
10-4172 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.07846 
.07875 
.07904 
.07933 
.07962 

.99692 
.99689 
.99687 
.99685 
.99683 

.07870 
.07899 
.07929 
.07958 
.07987 

12-7062 
12.6591 
12-6124 
12.5660 
12.5199 

.09585 
.09614 
.09642 
.09671 
.09700 

.99540 
.99537 
.99534 
.99531 
.99528 

.09629 
.09658 
.09688 
-09717 
-09746 

10-3854 
10-3538 
10-3224 
10.2913 
10-2602 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

.07991 
.08020 
.08049 
.08078 
.08107 

.99680 
.99678 
.99676 

.99673 
-99671 

.08017 
.08046 
.08075 
.08104 
-08134 

12-4742 
12-4288 
12-3838 
12-3590 

12-2946 

.09729 
.09758 
-09787 
.09816 
-09845 

.99526 
.99523 
-99520 
.99517 
.99514 

.09776 
-09805 
-09834 
.09864 
.09893 

10.2294 
10-1988 
10-1683 
xO.1381 

10-1080 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

.08136 
•  08165 
-08194 
.08223 
.08252 

.99668 

.99666 
.99664 
.99661 
.99659 

.08163 
.08192 
.08221 
.08251 
.08280 

12-2505 
12.2067 
12-1632 
12.1201 
12-0772 

-09874 
•09903 
.09932 
•09961 
.09990 

-99511 
-99508 
-99506 
.99503 
-99500 

.09923 
.09952 
-09981 
.10011 
.10040 

10-0780 
10-0483 
10  0187 
9-98931 
9.96007 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

•08281 
.08310 
.08339 
.08368 
.08397 

.99657 
.99654 
.99652 
.99649 
.99647 

.08309 
.08339 
.08368 
.08397 
.08427 

12-0346 
11-9923 
11-9504 
11-9087 
11-8673 

•10019 
•10048 
.10077 
.10106 
.10135 

.99497 
.99494 
.99491 
.99488 
-99485 

.10069 
.10099 
.10128 
.10158 
-10187 

9-93101 
9-90211 
9.87338 
9  •  84482 
_O184i_ 

15 
14 
13 
12 
_U 

50 

51 
52 
53 
54 

.08428 
.08455 
08484 
08513 
08542 

•99644 
.99642 
.99639 
.99637 
.99635 

.08456 
.08485 
.08514 
.08544 
.08573 

11.8262 
11.7853 
11.7448 
11.7045 
11.6645 

.10164 
.10192 
.10221 
.10250 
.10279 

-99482 
.99479 
.99476 
.99473 
.99470 

.10216 
.10246 
.10275 
.10305 
.10334 

9-78817 
9-76009 
9-73217 
9  •  70441 
9.67680 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

08571 
08600 
08829 
08658 
08687 

.99632 
.99630 
.99627 
.99625 
.99622 

.08602 
.08632 
.08661 
.08690 
.08720 

11-6248 
11-5853 
11.5461 
11.5072 
11.4685 

•10308 
.10337 
.10366 
-10395 
.10424 

.99467 
99464 
99461 
99458 
99455 

.10363 
10393 
10422 
10452 
10481 

9-64935 
9-62205 
9-59490 
9-56791 
9-54106 

5 
4 
3 
2 
1 

go 

08716 

.99619 

.08749 

11-4301 

.10453 

99452 

10510 

9-51436 

0 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot.  | 

Tan. 

85° 


84° 


NATURAL  SINES,   COSINES,   TANGENTS,   COTAN.     243 
6°  7° 


/ 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

/ 

0 

1 

2 
3 
4 

10453 
10482 
10511 
10540 
10569 

.99452 

•99449 
-99446 
.99443 
.99440 

10510 

10540 
10569 
10599 
10628 

9-51436 
9-48781 
9-46141 
9-43515 
9-40904 

12187 
12216 
12245 
12274 
12302 

99255 
99251 
99248 
99244 
99240 

.12278 
•12308 
•12338 
•12367 
•12397 

8-14435 
8-12481 
8-10536 
8-08600 
8-06674 

60 

59 
58 
57 
_5J 
55 
54 
53 
52 
51 
5O 
49 
48 
47 
46 
45 
44 
43 
42 

U 

5 
6 
7 
8 
9 
10 
11 
12 
13 
14 

10597 

10626 
10655 
.10684 
.10713 

.99437 
.99434 
.99431 
99428 
.99424 

10657 
10687 
10716 
10746 
10775 

9-38307 
9-35724 
9-33155 
9-30599 
9-28058 

12331 
12360 
12389 
12418 
12447 

99237 
99233 
99230 
99226 
99222 
99219 
99215 
99211 
.99208 
.99204 

•12426 
•12456 
12485 
12515 
.12544 

8-04756 
8-02848 
8-00948 
7-99058 
7.97176 
7-95302 
7-93438 
7-91582 
7-89734 
7-87895 
7-86064" 
7-84242 
7-82428 
7-80622 
7-78825 

.10742 
.10771 
.10800 
.10829 
.10858 

.99421 
.99418 
.99415 
.99412 
.99409_ 
.99406 
.99402 
.99399 
.9L396 
.99393 

10805 
10834 
10883 
10393 
_109_22_ 
10952 
10981 
11011 
11040 
.11070 

9-25530 
9  23016 
9-20516 
9.18028 
9.15554 
9-13093 
9-10646 
9.08211 
9-05789 
9-03379 

12476 
12504 
.12533 
.12562 
12591 

.12574 
.12603 
.12633 
.12662  . 
.12692^ 

15 
16 
17 
18 
19 
20 
21 
22 
23 
24 

,10887 
.10916 
.10945 
.10973 
.11002 

.12620 
.12649 
.12678 
.12706 
.12735 

•99200 
.99197 
.99193 
-99189 
-99186 

.12722 
.12751 
.12781 
.12810 
.12840 

.11031 
.11060 
.11089 
11118 
.11147 

.99390 
-99386 
.99383 
-99380 
.99377 

11099 
11128 
11158 
11187 
11217 

9-00983 
8-98598 
8-96227 
8-93867 
8.91520 
8-89185 
8-86862 
8-84551 
8-82252 
8-79964 

-12764 
-12793 
•12822 
.12851 
^12880 

.99182 
.99178 
-99175 
-99171 
.99167 

.12869 
.12899 
•12929 
.12958 
.12988 

7-77035 
7-7525-4 
7-73480 
7-71715 
7.69957 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

•  11176 
.11205 
11234 
.11263 
.11291 

.99374 
-93370 
.99367 
-99364 
.99360 

11246 
11276 
11305 
11335 
.11364 

.12908 
•12937 
.12968 
-12995 
.13024 

.99163 
99160 
-99156 
.99152 
.99148 

.13017 
.13047 
.13076 
.13106 
.13136 

7.68208 
7-66466 
7-64732 
7.63005 
7-61287 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.11320 
.11349 
.11378 
•11407 
-11436 

.99357 
.99354 
.99351 
.99347 
-99344_ 
.99341 
.99337 
•99334 
.99331 
.99327 

.11394 
11423 
.11452 
.11482 
.11511 

8.77689 
8-75425 
8-73172 
8-7093-1 
8-68701 

.13053 
.13081 
.13110 
.13139 
.13168 

.99144 
.99141 
.99137 
•99133 
.99129 

.13165 
.13195 

.13224 
.13254 
.13284 

7-59575 
7-57872 
7-56176 
7-54487 
7-52806 

30 

29 
28 
27 
28 
25 
24 
23 
22 
21 

35 
36 
37 
38 
39 

.11465 
.11494 
.11523 
-11552 
•  11580 

.11541 
.11570 
.11600 
.11629 
.11659 

8-66482 
8-64275 
8-62078 
8-59893 
8-57718 

.13197 
-13226 
.13254 
.13283 
.13312 

.99125 
.99122 
-993.18 
•99114 
-99110 

.13313 
-13343 
.13372 
.13402 
.13432 

7-51132 
7-49465 
7-47806 
7-46154 
7-44509 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49 

•11609 
.11638 
11667 
-11696 
.11725 

.99324 
-99320 
.99317 
.99314 
.99310 

.11688 
.11718 
.11747 
-11777 
-11806 

8.55555 
8-53402 
8-51259 
8.49128 
8-47007 

.13341 
.13370 
.13399 
-13427 
-13456 

-99106 
-90102 
-99098 
.99094 
.99091 

.13461 
.13491 
.13521 
.13550 
±13580_ 
.13609 
.13639 
.13669 
.13698 
.13728 

7-42871 
7-41240 
7.39616 
7-37999 
7.36389 

2O 

19 
18 
17 
16 
15 
14 
13 
12 
11 

-11754 
.11783 
.11812 
-11840 
-11869 

.99307 
•99303 
.99300 
.99297 
•99293 

.11836 
.11865 
.11895 
•11924 
.11954 
'.1  1983" 
,12013 
.12042 
•12072 
.12101 

8-44896 
8.42795 
8-40705 
8-38625 
8-36555 

-13485 
.13514 
.13543 
.13572 
.13600 

•99087 
.99083 
-99079 
-99075 
-99071 

7-34786 
7-33190 
7-31600 
7-30018 
7-28442 

50 

51 
52 
53 
54_ 
55 
56 
57 
58 
59 

.11898 
.11927 
.11956 
.11985 
-12014 

•99290 
.99286 
99283 
.99279 
.99276 

8.34496 
8-32446 
8-30406 
8.28376 
8-26355 

.13629 
.13658 
.13687 
•13716 
.13744 

.99067 
-99063 
.99059 
.99055 
.99051 

•13758 
.13787 
.13817 
.13846 
.13876 

7-26873 

7-25310 
7-23754 
7-22204 
7-20661 

10 

9 
8 
7 
6 
5 

3 
2 
1 

o 

•12043 
-12071 
.12100 
•12129 
.12158 

.99272 
.99269 
.99265 
.99262 
.99258 

.12131 
.12160 
•12190 
•12219 
.12249 

8  .  24345 
8-22344 
8-20352 
8-18370 
8.16398_ 

.13773 
.13802 
•13831 
-13860 
-13889 

.99047 
•99043 
99039 
•99035 
•99031 

.13906 
.13935 
.13965 
.13995 
.14024 

7-19125 
7-17594 
7.16071 
7-14553 
7-13042 

S2. 

.12187 

.99255 

12278 

8  14435 

.13917 

•99027 

•14054 

7-11537 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

8b° 


NATURAL  SINES,   COSINES,   TANGENTS,   COTAN. 
8°  9° 


1 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

» 

o 

2 

13917 
13946 
13975 
14004 
14033 

.99027 
.99023 
.99019 
.99015 
.99011 

14054 
14084 
14113 
14143 
14173 

7.11537 
7-10038 
7-08546 
7-07059 
7-05579 

15643 
15672 
15701 
15730 
15758 

•98769 
.98764 
-98760 
-98755 
-98751 

15838 
15868 
.15898 
•15928 
.15958 

6-31375 
6-30189 
6-29007 
6-27829 
6.26655 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

14061 
14090 
14119 
14148 
14177 

.99006 
.99002 
.98998 
.98994 
.98990 

14202 
14232 
14262 
14291 
14321 

7-04105 
7-02637 
7-01174 
6-99718 
6-98268 

15787 
15816 
15845 
15873 
15902 

-98746 
-98741 
.98737 
•98732 
-98728 

.15988 
.16017 
.16047 
.16077 
.16107 

6-25486 
6-24321 
6-23160 
6-22003 
6-20851 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

14205 
14234 
14263 
14292 
14320 

.98986 
.98982 
98978 
98973 
98969 

14351 
14381 
14410 
14440 
.14470 

6-96823 
6-95385 
6-93952 
6-92525 
6-91104 

15931 
15959 
15988 
16017 
16046 

-98723 
-98718 
-98714 
-98709 
-98704 

.16137 
.16167 
.16196 
.16226 
.16256 

6-19703 
6-18559 
6-17419 
6-16283 
6-15151 

5O 

49 
48 
47 
46 

15 
16 
17 
18 
19 

14349 
.14378 
.14407 
.14436 
-14464 

98965 
98961 
98957 
.98953 
.98948 

•14499 
.14529 
.14559 
.14588 
.14618 

6-89688 
6-88278 
6-86874 
6-85475 
6-84082 

16074 
16103 
16132 
16160 
16189 

-98700 
-98695 
-98690 
.98686 
.98681 

.16286 
.16316 
.16346 
-16376 
.16405 

6-14023 
6-12899 
6-11779 
6-10664 

6-09552 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

.  14493 
.14522 
.14551 
.14580 
.14608 

.98944 
.98940 
.98936 

.98931 
.98927 

.14648 
.14678 
.14707 
.14737 
.14767 

6-82694 
6-81312 
6-79936 
6-78564 
6-77199 

-16218 
.16246 
.16275 
.16304 
.16333 

•98676 
•98671 
.98667 
.98662 
.98657 

.16435 
.16465 
-16495 
•16525 
.16555 

6-08444 
6-07340 
6-06240 
6-05143 
6-04051 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

.14637 
.14666 
.14695 
.14723 
.14752 

.98923 
.98919 
•98914 
.98910 
.98906 

.14796 
.14826 
.14856 
.14886 
.14915 

6-75838 
6-74483 
6-73133 
6-71789 
6-70450 

.16361 
.16390 
.16419 
.16447 
,16476 

.98652 
.98648 
.98643 
-98638 
.98633 

.16585 

.16615 
-16645 
.16674 
.16704 

6-02962 
6-01878 
6-00797 
5-99720 
5-98646 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.14781 
.14810 
.14838 
.14867 
.14896 

.98902 
.98897 
.98893 
.98889 
.98884 

.14945 
.14975 
.15005 
.15034 
.15064 

6-69116 
6-67787 
6-66463 
6-65144 
6-63831 

.16505 
.16533 
.16562 
.16591 
.16620 

.98629 
.98624 
.98619 
-98614 
-98609 

.16734 
.16764 
.16794 
.16824 
.16854 

5-97576 
5-96510 
5-95448 
5-94390 
5-93335 

30 

29 
28 
27 
26 

35 
36 
37 
38 

39 

.14925 
.14954 
.14982 
.15011 
.15040 

•98880 
.98876 
.98871 
.98867 
.98863 

.15094 
.15124 
-15153 
.15183 
.15213 

6-62523 
6-61219 
6-59921 
6-58627 
6-57339 

-16648 
.16677 
.16706 
•16734 
-16763 

.98604 
.98600 
-98595 
.98590 
.98585 

•16884 
.16914 
.16944 
.16974 
.17004 

5-92283 
5-91236 
5-90191 
5-89151 

5-  ,88114 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

.15069 
.15097 
.15126 
.15155 
.15184 

.98858 
.98854 
.98849 
.98845 
-98841 

•15243 
.15272 
•15302 
.15332 
.15362 

6.56055 
6.54777 
6.53503 
6-52234 
6-50970 

.16792 
.16820 
,16849 
.16878 
.16906 

.98580 

.98575 
.98570 
.98565 
.98561 

-17033 
•17063 
.17093 
.17123 
.17153 

5-87080 
5-86051 
5-85024 
5.84001 
5.82982 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

.15212 
.15241 
.15270 
.15299 
.15327 

•98836 
•98832 
•98827 
•98823 
.98818 

.15391 
.15421 
.15451 
.15481 
.15511 

6-49710 
6-48456 
6-47206 
6-45961 
6-44720 

.16935 

.16964 
-16992 
•17021 
•17050 

.98556 
.98551 
.98548 
-98541 
-98536 

.17183 
.17213 
.17243 
-17273 
-17303 

5.81966 
5-80953 
5-79944 
5-78938 
5-77936 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

.15356 
.15385 
.15414 
•15442 
.15471 

•98814 
.98809 

.98805 
.98800 
.98796 

.15540 
.15570 
.15600 
.15630 
.15660 

6-43484 
6-42253 
6.41026 
6-39804 
6-38587 

-17078 
-17107 
-17136 
•17164 
•17193 

.98531 
-98526 
•98521 
•98516 
.98511 

•17333 
•17363 
.17393 
.17423 
.17453 

5-76937 
5-75941 
5-74949 
5-73960 
5-72974 

10 

9 
8 
7 

6 

55 
56 
57 
58 
59 

.15500 
.15529 
.15557 
•15586 
.15615 

.98791 
.98787 
-98782 
-98778 
.98773 

.15689 
.15719 
.15749 
•15779 
.15809 

6-37374 
6-36165 
6-34961 
6-33761 
6-32566 

-17222 
-17250 
.17279 
.17308 
.17336 

.98506 
.98501 
.98496 
-98491 
.98486 

•17483 
.17513 
.17543 
•17573 
.17603 

5-71992 
5-71013 
5-70037 
5-69064 
5-68094 

5 
4 
3 
2 
_J 

60 

.15643 

•98769 

.15838 

6-31375 

.17365 

-98481 

-17633 

5-67128 

0 

' 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

' 

81e 


NATURAL  SINES,   COSINES,   TANGENTS,   COTAN.     245 

10°  11° 


1 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

' 

o 

2 
3 
4 

.17365 
•17393 
-17422 
•17451 
.17479 

.98481 

.98476 
.98471 
.98466 
.98461 

.17633 
.17663 
.17693 
.17723 
.17753 

5-67128 
5-66165 
5-65205 
5  •  64248 
5-63295 

-19081 
-19109 
-19138 
-19167 
-19195 

-98163 
•98157 
.98152 
.98146 
.98140 

19438 
-19468 
-19498 
-19529 
•19559 

5-14455 
5-13658 
5-12862 
5-12069 
5-11279 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

.17508 
.17537 
.17565 
.17594 
•17623 

.98455 
.98450 
.98445 
.98440 
.98435 

.17783 
.17813 
.17843 
.17873 
.17903 

5-62344 
5-61397 
5-60452 
5-59511 
5-58573 

-19224 
-19252 
-19281 
-19309 
-19338 

.98135 
-98129 
•98124 
•98118 
•98112 

•19589 
•19619 
•19649 
.19680 
.19710 

5-10490 
5-09704 
5-08921 
5-08139 
5-07360 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.17651 
.17680 
.17708 
-17737 
.17766 

.98430 
•98425 
.98420 
•98414 
.98409 

17933 
.17963 
.17993 
.18023 
.18053 

5-57638 
5-56706 
5.55777 
5-54851 
5-53927 

-19366 
•19395 
•19423 
-19452 
-19481 

•98107 
•98101 
-98096 
.98090 
.98084 

-19740 
•19770 
.19801 
•19831 
19861 

5-06584 
5-05809 
5-05037 
5-04267 
5-03499 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19., 

.17794 
.17823 
.17852 
•17880 

.17909 

.98404 
.98399 
.98394 
.98389 
.98383 

.18083 
.18113 
.18143 
.18173 
•18203 

5-53007 
5-52090 
5-51176 
5-50264 
5-49356 

•19509 
•19538 
-19566 
-19595 

-19623 

-98079 
.98073 
•98067 
-98061 
•98056 

.19891 
.19921 
.19952 
-19982 
-20012 

5-02734 
5-01971 
5-01210 
5-00451 
4-99695 

45 
44 
43 
42 
41 

30 

21 
22 
23 

24 

17937 
.17966 
.17995 
.18023 
.18052 

.98378 
.98373 
.98368 
.98362 
.98357 

.18233 
•18263 
.18293 
.18323 
.18353 

5-48451 
5-47548 
5-46648 
5-45751 
5-44857 

-19652 
.19680 
.19709 
-19737 
-19766 

.98050 
.98044 
.98039 
.98033 
.98027 

-20042 
-20073 
,20103 
-20133 
•20164 

4.98940 
4-98188 
4-97438 
4-96690 
4.95945 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

.18081 
.18109 
.18138 
.18166 
•18195 

.98352 
•98347 
.98341 
.98336 
.98331 

.18384 
.18414 
.  18444 
.18474 
.18504 

5-43966 
5-43077 
5-42192 
5-41309 
5  •  40429 

.19794 
-19823 
-19851 
.19880 
.19908 

.98021 
.98016 
-98010 
-98004 
-97998 

.20194 

.20224 
•20254 
-20285 
-20315 

4-95201 
4-94460 
4-93721 
4-92984 
4-92249 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.18224 
.18252 
.18281 
.18309 
.18338 

•98325 
•98320 
.98315 
•98310 
.98304 

.18534 
.18564 
.18594 
-18624 
.18654 

5-39552 
5-38677 
5-37805 
5-36936 
5-36070 

•19937 
.19965 
•19994 
.20022 
-20051 

-97992 
-97987 
-97981 
-97975 
-97969 

-20345 
-20376 
-  20406 
-20436 
-20466 

4-91516 
4-90785 
4-90056 
4-89330 
4-88605 

30 

29 
28 
27 
26 

35 

36 
37 
38 

£9_ 

.18367 
.18395 
.18424 
.18452 
.18481 

.98299 
.98294 
.98288 
•982a3 
.98277 

•18634 
.18714 
.18745 
.18775 
.18805 

5-35206 
5-34345 
5-33487 
5-32631 
5-31778 

-20079 
-20108 
-20136 
-20165 
-20193 

-97963 
-97958 
-97952 
-97946 
-97940 

-20497 
-20527 
-20557 
-20588 
.20618 

4-87882 
4-87162 
4-86444 
4-85727 
4  85013 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

.18509 
.18538 
•18567 
.18595 
.18624 

.98272 
.98267 
.98261 
.98256 
.98250 

•18835 
•18865 
.18895 
.18925 
.18955 

5-30928 
5-30080 
5-29235 
5-28393 
5-27553 

•20222 
•20250 
-20279 
•20307 
-20336 

-97934 
-97928 
-97922 
-97916 
-97910 

-20648 
•20679 
-20709 
-20739 
-20770 

4-84300 
4-83590 
4-82882 
4-82175 
4-81471 

20 

19 
18 

17 
16 

45 
46 
47 
48 
49 

•18652 
.18681 
.18710 
•18738 
.18767 

.98245 
•98240 
.98234 
•98229 
.98223 

.18986 
.19016 
.19046 
•19076 
.19106 

5-26715 
5-25880 
5-25048 
5-24218 
5.23391 

-20364 
-20393 
-20421 
-20450 
-20478 

-97905 
-97899 
-97893 
-97887 
-97881 

-20800 
-20830 
-20861 
.20391 
.20921 

4-80769 
4-80068 
4-79370 
4-78673 
4-77978 

15 
14 
13 
12 
_ii 

50 

51 
52 
53 
54 

-18795 
.18824 
.18852 
.18881 
.18910 

•98218 
•98212 
•98207 
•98201 
.98196 

.19136 
•19166 
.19197 
.19227 
.19257 

5-22566 
5-21744 
5-20925 
5-20107 
5-19293 

-20507 
•20535 
.20563 
-20592 
-20620 

-97875 
-97869 
-97863 
-97857 
-97851 

-20952 
•20982 
.21013 
.21043 
.21073 

4-77286 
4-76595 
4-75908 
4-75219 
4-74534 

10 

9 

7 
6 

55 
56 
57 
58 
59 

•18938 
.18967 
.18995 
•19024 
.19052 

.98190 
.98185 
.98179 
•98174 
•98168 

.19287 
.19317 
•19347 
•19378 
•19408 

5-18480 
5-17671 
5-16863 
5-16058 
5-15256 

-  20649 
•20677 
.20706 
-20734 
-20763 

-97845 
-97839 
-97833 
-97827 
-97821 

-21104 
-21134 
-21164 
-21195 
-21225 

4-73851 
4-73170 
4-72490 
4-71813 
4-71137 

5 

4 
3 

! 

60 

.19081 

.98163 

.19438 

5-14455 

-20791 

-97815 

-21256 

4-70463 

o 

T- 

Cos. 

Sin. 

Cot. 

Tan. 

Co's. 

Sin. 

Cot. 

Tan. 

, 

79° 


78° 


246    NATURAL  SINES,   COSINES,   TANGENTS,   COTAN. 
12°  13° 


/ 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

' 

0 

1 

2 
3 
4 

20791 
20820 
20848 
.20877 
.20905 

.97815 
.97809 
.97803 
.97797 
.97791 

.21256 
•21286 
•21316 
.21347 
.21377  i 

4-70463 
4-69791 
4-69121 
4-68452 
4-67786 

22495 
22523 
22552 
22580 
22608 

97437 
97430 
97424 
97417 
97411 

23087 
23117 
23148 
23179 
23209 

4-33148 
4-32573 
4-32001 
4-31430 
4-30860 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

.20933 
.20962 
.20990 
.21019 
•21047 

•97784 
.97778 
•97772 
.97766 
.97760 

.21408 
•21438 
.21469 
.21499 
•21529 

4-67121 
4-66458 
4-65797 
4-65138 
4  •  64480 

22637 
22665 
22693 
22722 
22750 

97404 
-97398 
.97391 
•97384 
.97378 

.23240 
23271 
23301 
-23332 
-23363 

4-30291 
4-29724 
4-29159 
4-28595 
4-28032 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.21076 
.21104 
.21132 
.21161 
•  21189 

.97754 
.97748 
.97742 
.97735 
•97729 

•21560 
.21590 
-21621 
.21651 
-21682 

4-63825 
4-63171 
4-62518 
4-61868 
4-61219 

22778 
.22807 
-22835 
•22863 
-22892 

•97371 
.97365 
-97358 
-97351 
•97345 

•23393 
-23424 
-23455 
-23485 
-23516 

4-27471 
4-26911 
4-26352 
4-25795 
4-25239 

50 

49 
48 
47 
46 

15 
16 
17 
18 

19 

.21218 
.21246 
.21275 
.21303 
-21331 

.97723 
.97717 
•97711 
•97705 
•97698 

•21712 

.21743 
.21773 
.21804 
.21834 

4-60572 
4-59927 
4-59283 
4-58641 
4-58001 

.22920 
.22948 
.22977 
-23005 
.23033 

-97338 
•97331 
.97325 
-97318 
.97311 

-23547 
-23578 
-23608 
-23639 

.23670 

4-24685 
4-24132 
4-23580 
4-23030 

4-22481 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

-21360 
.21388 
•21417 
•  21445 
.21474 

.97692 
.97686 
.97680 
.97673 
.97667 

.21864 
.21895 
.21925 
.21956 
.21986 

4-57363 
4-56726 
4-56091 
4-55458 
4-54826 

.23062 
.23090 
.23118 
.23146 
.23175 

-97304 
-97298 
-97291 
-97284 
-97278 

-23700 
-23731 
-23762 
-23793 
-23823 

4-21933 
4-21387 
4-20842 
4-20298 
4-L9756 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

.21502 
.21530 
.21559 
.21587 
.21616 

.97661 
.97655 
.97648 
.97642 
•97636 

.22017 
•22047 
.22078 
.22108 
.22139 

4-54196 
4-53568 
4-52941 
4-52316 
4-51693 

.23203 
.23231 
.23260 
•23288 
-23316 

-97271 
-97264 
.97257 
-97251 
.97244 

-23854 
-23885 
-23916 
.23946 
•23977 

4-19215 
4-18675 
4-18137 
4-17600 
4-17064 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.21644 
.21672 
•21701 
•21729 
.21758 

•97630 
.97623 
.97617 
.97611 
.97604 

.22169 
.22200 
.22231 
.22261 
.22292 

4-51071 
4-50451 
4-49832 
4-49215 
4-48600 

•23345 
-23373 
•23401 
-23429 
-23458 

-97237 
-97230 
.97223 
•97217 
.97210 

•  24008 
.24039 
•24069 
•24100 
•24131 

4-16530 
4-15997 
4-15465 
4-14934 
4-14405 

30 

29 
28 
27 
26 

35 
38 
37 
38 
39 

.21786 
•  21814 
.21843 
.21871 

•  21899 

.97598 
.97592 
.97585 
.97579 
-97573 

•22322 

.22353 
•22383 
.22414 

.22444 

4-47986 
4-47374 
4-46764 
4-46155 
4-45548 

23486 
-23514 
.23542 
•23571 
.23599 

-97203 
-97196 
.97189 
-97182 
-97176 

•24162 
.24193 
•24223 
•24254 
-24285 

4-13877 
4-13350 
4-12825 
4-12301 
4-11778 

25 
24 
23 
22 
-21 

40 

41 
42 
43 
44 

.21928 
.21956 
.21985 
•22013 
•22041 

.97566 
-97560 
.97553 
.97547 
.97541 

.22475 
.22505 
•22536 
.22567 
.22597 

4-44942 
4-44338 
4-43735 
4-43134 

4-42534 

•23627 
.23656 
.23684 
•23712 
•23740 

-97169 
.97162 
.97155 
.97148 
.97141 

-24316 
-24347 
-24377 
-  24408 
-24439 

4-11256 
4-10736 
4-10216 
4-09699 
4-09182 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

•22070 
.22098 
•22126 
.22155 
•22183 

.97534 
.97528 
.97521 
.97515 
.97508 

.22628 
.22658 
.22689 
•22719 
•22750 

4-41936 
4-41340 
4-40745 
4-40152 
4-39560 

•23769 
•23797 
-23825 
-23853 
•23882 

.97134 
.97127 
•97120 
.97113 
.97106 

.24470 
.24501 
-24532 
-24562 
-24593 

4-08666 
4-08152 
4-07639 
4-07127 
4-06616 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

•22212 
•22240 
•22268 
.22297 
.22325 

.97502 
.97496 
.97489 
.97483 
.97476 

•22781 
•22811 
.22842 
.22872 
•2290S 

4-38969 
4-38381 
4-37793 
4-37207 
4-36623 

-23910 
•23938 
•23966 
-23995 
.24023 

.97100 
.97093 
.97086 
.97079 
.97072 

•24624 
-24655 
•24686 
•24717 
•24747 

4-06107 
4-05599 
4-05092 
4-04586 
4-04081 

10 

9 
8 
7 
_6 

55 
56 
57 
58 
59 

.22353 
.22382 
•22410 
•22438 
.22467 

.97470 
•97463 
.97457 
•97450 
.97444 

•22934 
•22964 
•22995 
.23026 
•23056 

4-36040 
4-35459 
4-34879 
4-34300 
4-33723 

.24051 
•24079 
•24108 
.24136 
.24164 

.97065 
-97058 
.97051 
.97044 
.97037 

•24778 
•24809 
•24840 
•24871 
-  24902 

4-03578 
4-03076 
4-02574 
4-02074 
4-01576 

5 
4 
3 
2 
1 

60 

•22495 

•97437 

.  23087 

4-33148 

•24192 

•97030 

-  24933 

4-01078 

0 

"  7— 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

. 

77* 


76° 


NATURAL  SINES,   COSINES,   TANGENTS,   COTAN.     247 

14°  15° 


< 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

0 

1 

2 
3 
4 

.24192 
.  24220 
.  24249 
.24277 
.24305 

.97030 
•97023 
.97015 
-97008 
.97001 

•24933 
•24964 
.24995 
•25026 
•25056 

4-01078 
4-00582 
4-00086 
3-99592 
3-99099 

-25882 
-25910 
-25938 
-25966 
-25994 

•96593 
-96585 
•96578 
•96570 
•96562 

-26795 
-26826 
•26857 
-26888 
•26920 

3-73205 
3-72771 
3-72338 
3-71907 
3-71476 

o;: 
5S 

58 
57 
56 

5 
6 
7 
8 
9 

.24333 
-24362 
-24390 
-24418 
-  24446 

•96994 
.96987 
.96980 
.96973 
-96966 

•25087 
•25118 
•25149 
•25180 

•25211 

3-98607 
3-98117 
3-97627 
3-97139 
3-96651 

•26022 
-26050 
-26079 
•26107 
•26135 

•96555 
•96547 
•96540 
-96532 
•96524 

•26951 
•26982 
•27013 
•27044 
•27076 

3-71046 
3-70616 
3-70188 
3-69761 
3-69335 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.  24474 
.24503 
-24531 
.24559 
.24587 

-96959 
-96952 
.96945 
.96937 
.96930 

•25242 
25273 
•25304 
•25335 
•25366 

3-96165 
3-95680 
3-95196 
3-94713 
3-94232 

•26163 
.26191 
•26219 
.26247 
•26275 

•96517 
•96509 
•96502 
•96494 
•96486 

•27107 
•27138 
•27169 
•27201 
•27232 

3-68909 
3-68485 
3-68061 
3-67638 
3-67217 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

.24615 
•  24644 
-24672 
.24700 

.24728 

.96923 
.96916 
-96909 
.96902 

.96894 

.25397 
•25428 
.25459 
.25490 

-25521 

3-93751 
3-93271 
3-92793 
3-92316 
3-91839 

-26303 
•26331 
.26359 
-26387 

.26415 

•96479 
•96471 
-96463 
•96456 

-96448 

•27263 
•27294 
•27326 
•27357 
•27388 

3-66796 
3-66376 
3-65957 
3-65538 
3-65121 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

•24756 
•24784 
.24813 
.24841 
.24869 

-96887 
-96880 
-96873 
-96866 
-96858 

-25552 
•25583 
•25614 
•25645 
•25676 

3-91364 
3-90890 
3-90417 
3  89945 
3-89474 

.28443 
.26471 
•26500 
-26528 
•26556 

-96440 
-96433 
•96425 
-96417 
•96410 

-27419 
•27451 
-27482 
•27513 
•27545 

3-64705 
3-64289 
3-63874 
3-63461 
3-63048 

4O 

39 
38 
37 
36 

25 
26 
27 
28 
29 

24897 
24925 
24954 
24982 
25010 

•96851 
-96844 
-96837 
-96829 
-96822 

•25707 
•25738 
•25769 
-25800 
•25831 

3-89004 
3-88536 
3-88068 
3-87601 
3-87136 

26584 
26612 
26640 
26668 
26696 

•96402 
-96394 
-96386 
•96379 
•96371 

•27576 
•27607 
•27638 
•27670 
•27701 

3-62636 
3-62224 
3-61814 
3-61405 
3-60996 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

25038 
25066 
25094 
25122 
25151 

96815 
96807 
96800 
96793 
96786 

.25862 
•25893 
.25924 
.25955 
-2598R 

3-86871 
3-86208 
3-85745 
3-85284 
3-84824 

26724 
26752 
26780 
26808 
26836 

-96363 
-96355 
-96347 
-96340 
•96332 

•27732 
•27764 
•27795 
•27826 
•27858 

3-60588 
3-60181 
3-59775 
3-59370 
3-58966 

30 

29 
28 
27 
26 

35 
38 
37 
38 
39_ 

25179 
25207 
25235 
25263 
25291 

96778 
96771 
96764 
96756 

96749 

•26017 
•26048 
•26079 
•26110 

-26141 

3-84364 
3-83906 
3-83449 
3-82992 
3-82537 

26864 
26892 
26920 
26948 

26976 

•96324 
•96316 
•96308 
•96301 

•96293 

•27889 
.27921 
-27952 
•27983 
28015 

3-58562 
3-58160 
3-57758 
3-57357 
3-56957 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

25320 
25348 
25376 
25404 
25432 

96742 
96734 
96727 
96719 
96712 

•26172 
•26203 
•28235 
•26266 
•26297 

3-82083 
3-81630 
3-81177 
3-80726 
3-80276 

27004 
27032 
27060 
27088 
27116 

•96285 
96277 
96269 
96261 
96253 

•  28046 
28077 
28109 
28140 
28172 

3-56557 
3-56159 
3-55761 
3-55364 
3-54968 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49_ 

25460 
25488 
25518 
25545 
25573 

96705 
96697 
96690 
96682 
96675 

•28328 
•26359 
•26390 
•26421 
•26452 

3-79827 
3-79378 
3-78931 
3-78485 
3  •  78040 

27144 
27172 
27200 
27228 
27256 

96246 
96238 
96230 
96222 
S6214 

28203 
28234 
28266 
28297 
28329 

3-54573 
3-54179 
3-53785 
3-53393 
3-53001 

15 
14 
13 
12 
11 

50 
51 
52 
53 
54 

25601 
25629 
25857 
25885 
25713 

96667 
96660 
96653 
96645 
96638 

.26483 
•26515 
•26546 
•26577 
•26608 

3-77595 
3-77152 
3-76709 
3-76268 
3-75828 

27284 
27312 
27340 
27368 
27396 

96206 
96198 
96190 
96182 
96174 

28360 
28391 
28423 
28454 
28486 

3-52609 
3-52219 
3-51829 
3-51441 
3-51053 

1O 

9 
8 
7 
6 

55 
56 
57 
56 
59 

25741 
25769 
25798 
25826 
25854 

96630 
96623 
96615 
96608 
•96600 

.26639 
.26670 
-26701 
•26733 
•26764 

3-75388 
3-74950 
3-74512 
3-74075 
3-73640 

27424 
27452 
27480 
27508 
27536 

96166 
96158 
96150 
96142 
96134 

28517 
28549 
28580 
28612 
28643 

3-50666 
3-50279 
3-49894 
3-49509 
3-49125 

5 
4 
3 
2 
1 

60 

25882 

-96593 

•26795 

3-73205 

27564 

96126 

28675 

3-48741 

0 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

' 

74° 


74° 


248    NATURAL  SINES,   COSINES,   TANGENTS,   COTAN. 

16°  17° 


/ 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

/ 

o 

1 

2 
3 
4 

•27564 
.27592 
.27620 
.27648 
.27676 

•96126 
.96118 
.96110 
.96102 
.96094 

•28675 
•28706 
•28738 
•28769 
.28800 

3-48741 
3-48359 
3-47977 
3-47596 
3-47216 

.29237 
•29265 
•29293 
.29321 
-29348 

•95630 
•95622 
.95613 
.95605 
-95596 

•30573 
•30605 
•30637 
•30669 
.30700 

3-27085 
3-26745 
3-26406 
3-26067 
3-25729 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

.27704 
.27731 
.27759 
.27787 
.27815 

.96086 
.96078 
.96070 
.96062 
.96054 

.28832 
•28864 
.28895 
.28927 
.28958 

3-46837 
3-46458 
3-46080 
3-45703 
3-45327 

.29376 
.29404 
.29432 
.29460 
-29487 

-95588 

.95579 
.95571 
.95562 
.95554 

•30732 
•30764 
.30796 
.30828 
.30860 

3-25392 
3-25055 
3-24719 
3-24383 
3  •  24049 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.27843 
.27871 
.27899 
.27927 
.27955 

.96046 
.96037 
.96029 

.96021 
.96013 

.28990 
.29021 
.29053 
.29084 
.29116 

3-44951 
3-44576 
3-44202 
3-43829 
3-43456 

-29515 
-29543 
.29571 
.29599 
.29626 

.95545 
.95536 
.95528 
.95519 
.95511 

-30891 
•30923 
-30955 
-30987 
-31019 

3-23714 
3-23381 
3-23048 
3-22715 
3-22384 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

.27983 
.28011 
.28039 
.28067 

-2809iL 

.96005 
.95997 
.95989 
.95981 
•95972 

.29147 
.29179 
.29210 
.29242 

.29274 

3-43084 
3-42713 
3-42343 
3-41973 
3-41604 

.29654 
.29682 
.29710 
•29737 
-29765 

.95502 
.95493 
.95485 
•95476 

-95467 

.31051 
.31083 
.31115 
.31147 
.31178 

3-22053 
3  21722 
3-21392 
3-21063 
3-20734 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

.28123 
.28150 
.28178 
.28206 
.28234 

.95964 
.95956 
.95948 
.95940 
.95931 

.29305 
.29337 
.29368 
.29400 
.29432 

3-41236 
3-40869 
3-40502 
3-40136 
3.39771 

.29793 
.29821 
-29C49 
-29876 
-29904 

.95459 
•954eO 
•9544i 
.95433 
.95424 

-31210 

.31243 
.31274 
•31306 
.31338 

3-20406 
3-20079 
3-19r/52 
3-19426 
3-19100 

4O 

39 
38 
37 
36 

25 
26 
27 
28 
29 

.28262 
.28290 
-28318 
.28346 
.28374 

.95923 
.95915 
.95907 
.95898 
.95890 

•29463 
.29495 
.29526 
.29558 
.29590 

3-39406 
3-39042 
3-38679 
3-38317 
3-37955 

.29932 
.29960 
.29987 
-30015 
.30043 

.95415 
.95407 
.95398 
.95389 
.95380 

•31370 
.31402 
•31434 
.31466 
.31498 

3-18775 
3-18451 
3-18127 
3-17804 
3.17481 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.28402 
.28429 
.28457 
.28485 
.28513 

.95882 
.95874 
.95865 
.95857 
.95849 

.29621 
.29653 
.29685 
.29716 
.29748 

3-37594 
3-37234 
3-36875 
3-36516 
3-36158 

-30071 

•30098 
.30126 
.30154 
.30182 

.95372 
.95363 
.95354 
.95345 
.95337 

.31530 
-31562 
-31594 
-31626 
.31658 

3-17159 
3-16838 
3-16517 
3-16197 
3-15877 

30 

29 
28 
27 
28 

35 
36 
37 
38 
39 

.28541 
•28569 
.28597 
.28625 
.28652 

.95841 

.95832 
.95824 
.95816 
.95807 

•29780 
.29811 
.29843 
.29875 
.2990fi_ 

3-35800 
3-35443 
3-35087 
3-34732 
3-34377 

.30209 
.30237 
.30265 
.30292 
.30320 

.95328 
.95319 
-95310 
-95301 

.95293 

.31690 
.31722 
.31754 
.31786 
.31818 

3-15558 
3-15240 
3-14922 
3-14605 
3-14288 

25 
24 
23 
22 
-21 

40 

41 
42 
43 
44 

.28680 
.28708 
.28736 
.28764 
.28792 

.95799 
.95791 
.95782 
.95774 
.95766 

.29938 
.29970 
.30001 
.30033 
.30065 

3-34023 
3-33670 
3-33317 
3-32965 
3-32614 

•30348 
•30376 
•30403 
.30431 
.30459 

-95284 
-95275 
-95266 
-95257 
-95248 

.31850 
.31882 
.31914 
.31946 
.31978 

3-13972 
3-13656 
3-13341 
3-13027 
3-12713 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

.28820 
28847 
•28875 
28903 
28931 

.95757 
•95749 
.95740 
.95732 
.95724 

.30097 
.30128 
.30160 
.30192 
.30224 

3-32264 
3-31914 
3-31565 
3-31216 
3-30868 

•30486 

.30514 
.30542 
.30570 
.30597 

-95240 

-95231 
-95222 
-95213 
-95204 

•32010 
.32042 
-32074 
.32106 
.32139 

3-12400 
3-12087 
3-11775 
3-11464 
3-11153 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

28959 
28987 
29015 
29042 
29070 

.95715 
•95707 
.95698 
•95690 
.95681 

30255 
30287 
30319 
30351 
30382 

3-30521 
3-30174 
3-29829 
3-29483 
3-29139 

•30625 
•30653 
.30680 
.30708 
.30736 

-95195 
.95186 
.95177 
.95168 
.95159 

32171 
-32203 
32235 
32267 
32299 

3-10842 
3-10532 
3-10223 
3-09914 
3-09606 

1O 

9 
8 
7 
6 

55 
56 
57 
58 
59 

29098 
29126 
29154 
29182 
29209 

.95673 
95664 
95656 
95647 
95639 

30414 
30446 
30478 
30509 
30541 

3-28795 
3-28452 
3.28109 
3-27767 
3-27426 

-30763 
-30791 
30819 
30846 
30874 

.95150 
•95142 
.95133 
.95124 
.95115 

32331 

32363 
32396 
32428 
32460 

3-09298 
3-08991 
3-08685 
3-08379 
3-08073 

5 

3 

2 
1 

go. 

29237 

95630 

30573 

3-27085 

30902 

•95106 

32492 

3-07768 

0 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

' 

73° 


78° 


NATURAL  SINES,   COSINES,   TANGENTS,   COTAN.    249 

18°  19° 


/ 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

/ 

0 

1 

2 
3 
4 

.30902 
.30929 
.30957 
.30985 
.31012 

.95106 
.95097 
.95088 
.95079 
.95070 

•32492 
•32524 
•32556 
•32588 
.32621 

3-07768 
3-07464 
3-07160 
3-06857 
3-06554 

•32557 
•32584 
•32612 
•32639 
•32667 

•94552 
•94542 
•94533 
-94523 
-94514 

.  34433 
•34465 
.34498 
-34530 
.34563 

2-90421 
2-90147 
2-89873 
2-89600 
2-89327 

6O 

59 
58 
57 
56 

5 
6 
7 
8 
9 

.31040 
.31068 
.31095 
.31123 
.31151 

.95061 
.95052 
.95043 
.95033 
.95024 

•32653 
.32685 
.32717 
.32749 
.32782 

3-06252 
3-05950 
3-05649 
3-05349 
3-05049 

.32694 
.32722 
•32749 
.32777 
.32804 

-94504 
.94495 
•94485 
.94476 
.94466 

.34596 

-34628 
.34661 
-34693 
-34726 

2-89055 
2-88783 
2-88511 
2-88240 
2-87970 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.31178 
.31206 
.31233 
.31261 
.31289 

.95015 
.95006 
.94997 
.94988 
.94979 

.32814 
.32846 
.32878 
.32911 
.32943 

3  •  04749 
3  •  04450 
3-04152 
3-03854 
3-03556 

.32832 
•32859 
•32887 
•32914 
.32942 

-94457 
.  94447 
-94438 
.94428 
-94418 

-34758 
.34791 
.34824 
.34856 
.34889 

2-87700 
2  •  87430 
2-87161 
2-86892 
2-86624 

5O 

49 
48 
47 
46 

15 
16 
17 
18 
19 

-31316 
.31344 
•31372 
.31399 

.31427 

.94970 
.94961 
.94952 
.94943 
.94933 

.32975 
.33007 
.33040 
.33072 

.33104 

3-03260 
3-02963 
3-02667 
3-02372 
3-02077 

.32969 
.32997 
.33024 
.33051 
.33079 

-  94409 
.94399 
-943SO 
.94380 
.94370 

-34922 
-34954 
-34987 
.35020 
.35052 

2-86356 
2-86089 
2-85822 
2-85555 
2-85289 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

.31454 
.31482 
.31510 
.31537 
-31565 

.94924 
.94915 
.94906 
.94897 
.94888 

-33136 
-33169 
-33201 
.33233 
.33266 

3-01783 
3-01489 
3-01196 
3-00903 
3-00611 

-33106 
.33134 
.33161 
.33189 
.33216 

-94361 
-94351 
-94342 
.94332 
.94322 

.35085 
.35118 
•35150 
.35183 
-35216 

2-85023 
2-84758 
2-84494 
2-84229 
2-83965 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

•31593 
•31620 
-31648 
•31675 
.31703 

.94878 
.94869 
•94860 
•94851 
.94842 

-33298 
-33330 
.33363 
.33395 
.33427 

3-00319 
3-00028 
2-99738 
2-99447 
2-99158 

•33244 
-33271 
33298 
.33326 
-33353 

-94313 
.94303 
.94293 
.94284 
.94274 

.35248 
.35281 
.35314 
•35346 
.35379 

2-83702 
2-83439 
2-83176 
2-82914 
2-82653 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

•31730 
.31758 
.31786 
.31813 
•31841 

.94832 
.94823 
.94814 
•94805 
.94795 

•33460 
•33492 
•33524 
•33557 
•33589 

2-98868 
2-98580 
2-98292 
2-98004 
2-97717 

.33381 

.33408 
•33436 
•33463 
•33490 

.94264 
.94254 
.94245 
.94235 
.94225 

.35412 
.35445 
.35477 
.35510 
.35543 

2-82391 
2-82130 
2-81870 
2-81610 
2-81350 

30 

29 
28 
27 
38 

35 
36 
37 
38 

39 

•31868 
.31896 
•31923 
.31951 

.31979 

.94786 
.94777 
.94768 
.94758 
.94749 

•33621 
.33654 
-33686 
-33718 

-33751 

2-97430 
2-97144 
2-96858 
2-96573 
2.96288 

-33518 

•33545 
.33573 
•33600 
.33627 

-94215 
•94206 
.94196 
•94186 
.94176 

-35576 
-35608 
•35641 
.35674 
.35707 

2-81091 
2-80833 
2-80574 
2-80316 
2  80059 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

.32006 
•32034 
.32061 
•32089 
•321i6 

.94740 
.94730 
•94721 
.94712 
•94702 

.33783 
•33816 
•33848 
•33381 
.33913 

2-96004 
2-95721 
2-95437 
2-95155 
2-94872 

.33655 
•33682 
•33710 
•33737 
•33764 

-94167 
-94157 
-94147 
-94137 
-94127 

-35740 
-35772 
.35805 
.35838 
•35871 

2-79802 
2-79545 
2-79289 
2-79033 
2-78778 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

.32144 
.32171 
.32199 
.32227 
.32254 

.94693 
.94684 
.94674 
•94865 
-94856 

.33945 
•33978 
.34010 
34043 
.34075 

2-94591 
2-94309 
2-94028 
2-93748 
2-93468 

•33792 
•33819 
•33846 
•33874 
•33901 

-94118 
•94108 
.94098 
-94088 
-94078 

.35904 
.35937 
.35969 
.36002 
.36035 

2-78523 
2-78269 
2-78014 
2-77761 
2-77507 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

.32282 
.32309 
.32337 
.32364 
.32392 

.94646 
•94637 
.94627 
.94618 
.94609 

•34108 
.34140 
•34173 
.34205 
.34238 

2-93189 
2-92910 
2-92632 
2-92354 
2-92076 

-33929 
•33956 
•33983 
-34011 
•34038 

-94068 
•94058 
.  94049 
-94039 
•94029 

.36068 
.36101 
.36134 
•36167 
.36199 

2-77254 
2-77002 
2-76750 
2-76498 
2-76247 

10 

9 

7 
6 

55 
56 
57 
58 
59 

•32419 
.32447 
•32474 
•32502 
-32529 

•94599 
.94590 
.94580 
•94571 
.94561 

.34270 
.34303 
.34335 
•34368 
.  34400 

2-91799 
2-91523 
2-91246 
2-90971 
2-90696 

•34065 
•34093 
•34120 
•34147 
•34175 

-94019 
.94009 
-93999 
•93989 
-93979 

•36232 
.36265 
.36298 
.36331 
.36364 

2-75996 
2-75746 
2-75496 
2-75246 
2-74997 

5 
4 
3 
2 

60 

.32557 

•94552 

.34433 

2.90421 

-34202 

.93969 

.38397 

2-74748 

Q 

' 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

71° 


70° 


250    NATURAL  SINES,   COSINES,   TANGENTS,   COTAN. 

20°  21° 


/ 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

/ 

0 

1 

2 
3 

4 

.34202 

.34229 
•34257 
•34284 
.34311 

•93969 
•93959 
•93949 
.93939 
-93929 

•36397 
.36430 
•36463 
•36496 
•36529 

2-74748 
2-74499 
2-74251 
2-74004 
2-73756 

-35837 

-35864 
-35891 
.35918 
-35945 

.93358 
-93348 
-93337 
-93327 
-93316 

•38386 

•38420 
-38453 
.38487 
.38520 

2-60509 
2-60283 
2-60057 
2-59831 
2-59606 

6O 

59 
58 
57 
56 

5 
6 
7 
8 
9 

.34339 
.34366 
•34393 
.  34421 
.34448 

•93919 
.93909 
.93899 
.93889 
.93879 

•36562 
•36595 
•36628 
•36661 
.36694 

2-73509 
2-73263 
2-73017 
2-72771 
2-72526 

-35973 
•36000 
-36027 
.36054 
.36081 

-93306 
-93295 
.93285 
•93274 
•93264 

.38553 
.38587 
•38620 
.38654 
.38687 

2-59381 
2-59156 
2-58932 
2-58708 
2-58484 

55 
54 
53 
52 
51 

10 

11 

12 
13 
14 

•34475 
•34503 
.34530 
.34557 
.34584 

.93869 
.93859 
.93849 
•93839 
.93829 

-36727 
-36760 
.36793 
.36826 
.36859 

2-72281 
2-72036 
2-71792 
2-71548 
2-71305 

•36108 
.36135 
•36162 
.36190 
.36217 

.93253 
•93243 
.93232 
•93222 
-93211 

-38721 
•38754 
.38787 
•38821 
.38854 

2-58261 
2-58038 
2-57815 
2-57593 
2-57371 

50 

49 
48 

47 
4B 

15 
16 
17 
18 
i?_ 

.34612 
34639 
.34666 
.34694 
•34721 

.93819 
.93809 
•93799 
.93789 
.93779 

•36892 

.36925 
.36958 
.36991 
-37024 

2-71062 
2-70819 
2-70577 
2-70335 
2-70094 

.36244 
-36271 
-36298 
.36325 
.36352 

-93201 
-93190 
-93180 
-93169 
-93159 

.38888 
.38921 
.38955 
.38988 
•39022 

2-57150 
2-56928 
2-56707 
2-56487 
2-56266 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

-34748 
.34775 
.34803 
.34830 
.34857 

-93769 
•93759 
•93748 
•93738 
.93728 

-37057 
.37090 
-37123 
.37157 
-37190 

2-69853 
2-69612 
2-69371 
2-69131 
2-68892 

•36379 
.36406 
-36434 
-36461 
-36488 

-93148 
-93137 
-93127 
-93116 
-93106 

•39055 
•39089 
.39122 
•39156 
•39190 

2-56046 
2-55827 
2-55608 
2-55389 
2-55170 

40 

39 
38 
37 
SB 

25 
26 
27 
28 
29 

•34884 
-34912 
•34939 
.34966 
.34993 

•93718 
•93708 
.93698 
•93688 
.93677 

-37223 
•37256 
.37289 
.37322 
.37355 

2-68653 
2-68414 
2.68175 
2-67937 
2.67700 

-36515 
•36542 
.36569 
.38596 
.36623 

•93095 

-93084 
.93074 
•93063 
.93052 

-39223 
39257 
•39290 
.39324 
•39357 

2-54952 
2-54734 
2.54516 
2.54299 
2-54082 

35 
34 
33 
32 
31 

30 

31 

32 
33 
34 

.35021 
.35048 
-35075 
-35102 
-35130 

.93667 
.93657 
•93647 
.93637 
•93626 

•37388 

.37422 
•37455 
.37488 
.37521 

2.67462 
2.67225 
2-66989 
2-66752 
2-66516 

•36650 
.36677 
.36704 
.36731 
.36758 

•93042 
•93031 
•93020 
•93010 
.92999 

39391 
-39425 
-39458 
-39492 
-39526 

2-53865 
2.53648 
2-53432 
2-53217 
2-53001 

3O 

29 
28 
27 
26 

35 
36 
37 
38 
39 

-35157 
.35184 
.35211 
•35239 
-35266 

•93616 
•93606 
•93596 
•93585 
•93575 

•37554 
-37588 
•37621 
.37654 
.37687 

2-66281 
2-66046 
2-65811 
2-65576 
2-65342 

•36785 
.36812 
.36839 
.36867 
.36894 

•92988 
•92978 
•92967 
•92956 

.92945 

•39559 
.39593 
•39626 
•39660 
.39694 

2-52786 
2-52571 
2-52357 
2-52142 
2-51929 

25 
24 
23 
22 
21 

4O 

41 
42 
43 
44 

-35293 
-35320 
-35347 
-35375 
-35402 

.93565 
•93555 
•93544 
.93534 
.93524 

-37720 
•37754 
•37787 
.37820 
.37853 

2-65109 
2-64875 
2-64642 
2-64410 
2-64177 

•36S21 
•36948 
-36975 
-37002 
-37029 

.92935 
•92924 
-92913 
-92902 
-92892 

-39727 
.39761 
•39795 
.39829 
-39862 

2-51715 
2-51502 
2-51289 
2-51076 
2-50864 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

-35429 
•35456 
.35484 
-35511 
-35538 

•93514 
•93503 
•93493 
.93483 
•93472 

-37887 
.37920 
•37953 
•37986 
.38020 

2-63945 
2-63714 
2-63483 
2-63252 
2-63021 

.37056 
-37083 
-37110 
.37137 
-37164 

.92881 
•92870 
•92859 
•92849 
.92838 

•39896 
.39930 
-39963 
•39997 
.40031 

2-50652 
2-50440 
2-50229 
2-50018 
2-49807 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

•35565 
.35592 
•35619 
.35647 
•35674 

•93462 
•93452 
•93441 
-93431 
•93420 

-38053 
-38086 
.38120 
•38153 
.38186 

2-62791 
2-62561 
2.62332 
2-62103 
2-61874 

-37191 
-37218 
-  37245 
•37272 
.37299 

•92827 
-92816 
-92805 
-92794 
-92784 

.40065 
•40098 
.40132 
.40166 
-  40200 

2-49597 
2-49386 
2-49177 
2-48967 
2-48758 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

•35701 
35728 
35755 
35782 
35810 

.93410 
.93400 
•93389 
.93379 
•93368 

-38220 
.38253 
•38286 
.38320 
.38353 

2.61646 
2-61418 
2-61190 
2.60963 
2-60736 

.37326 
.37353 
•37380 
•37407 
-37434 

-92773 
-92762 
.92751 
•92740 
•92729 

40234 
40267 
40301 
40335 
40369 

2-48549 
2-48340 
2-48132 
2-47924 
2.47716 

5 
4 
3 
2 
1 

60 

35837 

•93358 

•38386 

2-60509 

-37461 

92718 

40403 

2-47509 

0 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

' 

69° 


68° 


NATURAL  SINES,   COSINES,   TANGENTS,   COTAN.     251 
33°  33° 


1 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

f 

0 

1 

2 
3 

•37461 
.37488 
.37515 
.37542 
.37569 

•92718 
.92707 
•92697 
•92686 
•92675 

.  40403 
.40436 
.  40470 
.40504 
.40538 

2-47509 
2-47302 
2.47095 
2.46888 
2-46682 

•39073 
•39100 
•39127 
•39153 
.39180 

•92050 
•92039 
.92028 
•92016 
.92005 

•42447 
•42482 
•42516 
•42551 
•42585 

2-35585 
2-35395 
2-35205 
2-35015 
2-34825 

60 

59 
58 
57 
5fi 

5 
6 
7 
8 
9 

.37595 
.37622 
.37649 
.37676 
.37703 

•92664 
.92653 
.92642 
.92631 
92620 

.40572 

•40606 
.40640 
.40674 
.40707 

2-46476 
2-46270 
2-46065 
2-45860 
2-45655 

•39207 
•39234 
•39280 
.39287 
-39314 

.91994 
•91982 
.91971 
.91959 
-91948 

•42619 
•42654 
•42688 
.42722 
.42757 

2-34636 
2-34447 
2-34258 
2-34069 
2-33881 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.37730 
.37757 
•37784 
.37811 
.37838 

.92609 
.92598 
.92587 
.92576 
.92565 

•40741 
.40775 
•40809 
•40843 
.40877 

2-45451 
2-45246 
2-45043 
2-44839 
2.44636 

-39341 
-39367 
-39394 
-39421 
-  39448 

-91936 
.91925 
.91914 
.91902 
.91891 

•42791 
.42826 
•42860 
.42894 
.42929 

2-33693 
2-33505 
2-33317 
2-33130 
2-32943 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19  , 

•37865 
.37892 
.37919 
.37946 
.37973 

.92554 
.92543 
•92532 
.92521 

.92510 

.40911 
•40945 
.40979 
.41013 

.41047 

2-44433 
2-44230 
2-44027 
2-43825 
2-43623 

-39474 
-39501 
-39528 
-39555 
.39581 

.91879 
•91868 
•91856 
•91845 
.91833 

•42963 
.42998 
•43032 
•43067 
•43101 

2-32756 
2-32570 
2-32383 
2-32197 
2-32012 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

.37999 
.38026 
•38053 
.38080 
.38107 

•92499 
•92488 
.92477 
•92466 
•92455 

.41081 
.41115 
.41149 
.41183 
.41217 

2-43422 
2-43220 
2-43019 
2-42819 
2-42618 

-39608 
-39635 
-39661 
.39688 
-39715 

•91822 
.91810 
.91799 
•91787 
.91775 

•43136 
•43170 
.43205 
.43239 
.43274 

2-31826 
2-31641 
2-31456 
2-31271 
2-31086 

40 

39 
38 
37 
3« 

25 
26 
27 
28 
29 

.38134 
.38161 
.38188 
-38215 
.38241 

.92444 
.92432 
.92421 
.92410 
•92399 

.41251 
.41285 
•41319 
•41353 
.41387 

2-42418 
2-42218 
2-42019 
2-41819 
2-41620 

•39741 
.39768 
.39795 
.39822 
.39848 

•91764 
•91752 
•91741 
.91729 
-91718 

•43308 
.43343 
.43378 
•43412 
.43447 

2-30902 
2-30718 
2-30534 
2-30351 
2-30167 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.38268 
.38295 
.38322 
.33349 
.38376 

•92388 
.92377 
•92366 
•92355 
•92343 

.41421 
•41455 
•41490 
.41524 
.41558 

2  41421 
2-41223 
2-41025 
2-40827 
2-40629 

•39875 
.39902 
.39928 
-39955 
•39982 

.91706 
-91694 
-91683 
•91671 
•91660 

.43481 

.43516 
.43550 
•43585 
•43620 

2-29984 
2-29801 
2-29619 
2-29437 
2-29254 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

•38403 
.38430 
.38456 
.38483 
.38510 

.92332 
.92321 
•92310 
•92299 
.92287 

.41592 
•41626 
•41660 
•41694 
.41728 

2-40432 
2-40235 
2-40038 
2-39841 
2-39645 

-  40008 
.40035 
-40062 
-40088 
-40115 

•91648 
.91636 
.91625 
•91613 

-91601 

•43654 
.43689 
.43724 
.43758 
.43793 

2-29073 
2-28891 
2-28710 
2-28528 
2-28348 

25 
24 
23 
22 
21 

40 

41 
42 
43 

44 

•38537 

•38564 
.38591 
.38617 
-38644 

.92276 
.92265 
.92254 
.92243 
•92231 

.41763 
•41797 
•41831 
.41865 
.41899 

2-39449 
2-39253 
2-39058 
2-38863 
2-38668 

-40141 
-40168 
-40195 
•40221 
•  40248 

.91590 
•91578 
.91566 
.91555 
.91543 

•43828 

.43862 
•43897 
•43932 
.43966 

2-28167 
2-27987 
2-27806 
2-27626 
2  -  27447 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

-38671 
.38698 
-38725 
•38752 
.33778 

•92220 
•92209 
.92198 
.92186 
•92175 

.41933 
.41968 
.42002 
.42036 
.42070 

2-38473 
2-38279 
2-38084 
2-37891 
2-37697 

•40275 
40301 
.40328 
.40355 
.40381 

•91531 
•91519 
.91508 
.91496 
•91484 

•44001 
.44036 
.44071 
.44105 
•44140 

2-27267 
2-27088 
2-26909 
2-26730 
2-26552 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

•38805 
.38832 
.38859 
•38886 
.38912 

.92164 
•92152 
•92141 
•92130 
.92119 

.42105 
.42139 
.42173 
.42207 
•42242 

2-37504 
2-37311 
2-37118 
2-36925 
2-36733 

.  40408 
-40434 
-40461 
-40488 
-40514 

•91472 
•91461 
•91449 
•91437 
•91425 

•44175 
•44210 
.  44244 
.44279 
•44314 

2-26374 
2-26196 
2-26018 
2-25840 

1O 

9 
8 
7 
6 

55 
56 
57 
58 
59 

•38939 
•38966 
•38993 
.39020 
.39046 

.92107 
.92096 
•92085 
•92073 
.92062 

•42276 
.42310 
•42345 
•42379 
•42413 

2-36541 
2-36349 
2-36158 
2-35967 
2-35776 

.40541 
•40567 
•40594 
•40621 
•40847 

•91414 
.91402 
.91390 
.91378 
•91366 

.44349 
•44384 
•44418 
•44453 
.  44488 

2-25486 
2-25309 
2-25132 
2-24956 
2-24780 

5 

3 
2 
1 

60 

-39073 

.92050 

.  42447 

2-35585 

•40674 

•91355 

-44523 

2  •  24604 

_o 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

f 

67° 


66° 


252    NATURAL  SINES,   COSINES,   TANGENTS,   COTAN. 
24*  25° 


t 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

/ 

o 

1 

2 
3 

•40674 
.40700 
•40727 
.40753 
.40780 

•91355 
•91343 
.91331 
.91319 
•9i307 

•44523 
•44558 
-44593 
.44627 
.44662 

2-24604 
2-24428 
2-24252 
2-24077 
2.23902 

•42262 
.42288 
.42315 
-42341 
-42367 

-90631 
-90618 
-90606 
•90594 
•90o82 

-46631 
•46666 
•46702 
•46737 
•46772 

2-14451 
2-14288 
2-14125 
2-13963 
2.13801 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

.40806 
.40833 
.40860 
.40886 
.40913 

•91295 
.91283 
.91272 
•91260 
.91248 

•44697 
•44732 
•44767 
•44802 
•44837 

2-23727 
?.  23553 
2.23378 
2-23204 
2-23030 

•42394 
-42420 
-42446 
•42473 
-42499 

-90569 
-90557 
•90545 
•90532 
•90520 

.46808 
.46843 
•46879 
•46914 
•46950 

2-13639 
2-13477 
2-13316 
2-13154 
2-12993 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

•40939 
•40966 
•40992 
•41019 
.41045 

.91236 
.91224 
.91212 
.91200 
.91188 

•44872 
•44907 
•44942 
•44977 
•45012 

2-22857 
2-22683 
2.22510 
2-22337 
2-22164 

-42525 
-42552 
-42578 
•42604 
.42631 

•90507 
•90495 
.90483 
-  90470 
-90458 

•46985 
.47021 
.47056 
•47092 
•47128 

2-12832 
2-12671 
2-12511 
2-12350 
2-12190 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

.41072 
.41098 
.41125 
.41151 
.41178 

•91176 
•91164 
•91152 
.91140 
.91128 

•45047 
•45082 
45117 

•45152 
.45187 

2.21992 
2-21819 
2-21647 
2-21475 
2-21304 

.42657 
.42683 
.42709 
•42736 
•42762 

-90446 
-90433 
•90421 
-90408 
•90396 

.47163 
.47199 
.47234 
.47270 
-47305 

2-12030 
2-11871 
2.11711 
2-11552 

2-11392 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

.41204 
•41231 
•41257 
•41284 
•41310 

.91116 
•91104 
.91092 
.91080 
91068 

-45222 
•45257 
-45292 
-45327 
-45362 

2-21132 
2-20961 
2-20790 
2-20619 
2-20449 

•42788 

.42815 
.42841 
•42867 
-42894 

.90383 
•90371 
-90358 
-90346 
-90334 

-47341 
.47377 
.47412 
.47448 
-47483 

2-11233 
2.11075 
2-10916 
2-10758 
2-10600 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

•41337 
•41363 
•41390 
•41416 
•41443 

-91056 
-91044 
-91032 
.91020 
•91008 

•45397 

•45432 
•45467 
•45502 
.45538 

2-20278 
2-20108 
2.19938 
2-19769 
2.19599 

.42920 
•42946 
•42972 
-42999 
-43025 

•90321 
•90309 
•90296 
•90284 
•90271 

.47519 
-47555 
.47590 
.47626 
.47662 

2  •  10442 
2-10284 
2-10126 
2-09969 
2-09811 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

•41469 
•41496 
•41522 
•41549 
•41575 

.90996 
•90984 
•90972 
.90960 
.90948 

•45573 
-45608 
-45643 
-45678 
•45713 

2-19430 
2-19261 
2-19092 
2-18923 
2-18755 

•43051 
•43077 
-43104 
-43130 
-43156 

•90259 

•90246 
•90233 
•90221 
.90208 

.47698 
.7733 
.47769 
.47805 
•47840 

2-09354 
2-09498 
2-09341 
2-09184 
2-09028 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

•41602 
•41628 
•41655 
.41681 
.41707 

•90936 
-90924 
-90911 
•90899 
.90887 

•45748 
•45784 
•46819 
•45854 
•45889 

2-18587 
2-18419 
2-18251 
2-18084 
2-17916 

-43182 
-43209 
•43235 
-43261 

-43287 

•90196 
•90183 
•90171 
•90158 

.90146 

•47876 
•47912 
•47948 
•47984 
-48019 

2-08872 
2-08716 
2-08560 
2-08405 
2-08250 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

•41734 
.41760 
•41787 
•41813 
•41840 

.90875 
•90863 
•90851 
•90839 
-90826 

•45924 
•45960 
•45995 
•46030 
•46065 

2-17749 
2-17582 
2-17416 
2-17249 
2-17083 

-43313 
-43340 
43366 
-43392 
.43418 

.90133 
•90120 
.90108 
.90095 
.90082 

•48055 
•48091 
•48127 
•48163 
•48198 

2-08094 
2-07939 
2-07785 
2-07630 
2-07476 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

•41866 
•41892 
•41919 
•41945 
•41972 

•90814 
•90802 
•90790 
•90778 
•90766 

•46101 
•46136 
•46171 
•46206 
•46242 

2-16917 
2-16751 
2.16585 
2-16420 
2-16255 

-  43445 
•43471 
•43497 
•43523 
-43549 

•90070 
•90057 
90045 
•90032 
-90019 

-48234 
-48270 
-48306 
-48342 
-48378 

2.07321 
2-07167 
2-07014 
2-06860 
2-06706 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

•41998 
•42024 
42051 
•42077 
•42104 

•90753 
-90741 
-90729 
.90717 
90704 

•46277 
•46312 
-46348 
•46383 
•46418 

2-16090 
2-15925 
2-15760 
2-15596 
2-15432 

-43575 
-43602 
•43628 
•43654 
.43680 

-90007 
•89994 
•89981 
.89968 
.89956 

.48414 
.48450 
-48486 
-48521 
•48557 

2-06553 
2  -  06400 
2-06247 
2-06094 
2-05942 

10 

9 
8 

6 

55 
56 
57 
58 
59 

•42130 
•42156 
•42183 
•42209 
•42235 

.90692 
.90680 
.90668 
.90655 
.90643 

•46454 
.46489 
.46525 
•46560 
.46595 

2-15268 
2-15104 
2-14940 
2  14777 
2-14614 

•43706 
•43733 
-43759 
-43785 
-43811 

•89943 
-89930 
-89918 
•89905 
-89892 

•48593 
.48629 
-48665 
•48701 
-48737 

2-05790 
2-05637 
2-05485 
2-05333 
2-05182 

5 

3 
2 
1 

60 

.42262 

•90631 

•46631 

2-14451 

-43837 

-89879 

-48773 

2-05030 

_o 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

65° 


64° 


NATURAL  SINES,   COSINES,   TANGENTS,   COTAN.     253 
86°  37° 


t 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

/ 

0 

1 

2 
3 
4 

•43837 
•43863 
.43889 
.43916 
.43942 

•89879 
•89867 
.89854 
.89841 
.89828 

•48773 
.48809 
.48845 
.48881 
.48917 

2-05030 
2-04879 
2-04723 
2.04577 
2-04426 

•45399 
-45425 
.45451 
.45477 
.45503 

.89101 
.89087 
.89074 
.89061 
.89048 

-50953 
.50989 
-51026 
-51063 
-51099 

1-96261 
1.96120 
1-95979 
1.95838 
1.95698 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

•43968 
.43994 
.44020 
.  44046 
.44072 

.89816 
.89803 
.89790 
.89777 
.89764 

.48953 
.48989 
.49026 
.49062 
.49098 

2-04276 
2-04125 
2-03975 
2-03825 
2-03675 

.45529 
.45554 
.45580 
.45606 
-45632 

•89035 
-89021 
-89008 
-88995 
.88981 

51136 
.51173 
.51209 
.51246 
-51283 

1-95557 
1-95417 
1-95277 
1-95137 
1-94997 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.44098 
.44124 
.44151 
-44177 
.44203 

.89752 
.89739 
.89726 
.89713 
.89700 

.49134 
.49170 
•49206 
•49242 
.49278 

2-03526 
2-03376 
2-03227 
2-03078 
2-02929 

•45658 
.45684 
.45710 
.45736 
.45762 

-88968 
.88955 
.88942 
-88928 
.88915 

-51319 
-51356 
-51393 
.51430 
•51467 

1-94858 
1-94718 
1.94579 
1  .  94440 
1-94301 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

•44229 
•44255 
.44281 
.44307 
.44333 

.89687 
.89674 
.89662 
.89649 
-89636 

.49315 
.49351 
.49387 
.49423 
.49459 

2-02780 
2-02631 
2-02483 
2-02335 
2-02187 

•45787 
.45813 
.45839 
.45865 
.45891 

-88902 
-88888 
-88875 
.88862 
.88848 

.51503 

-51540 
-51577 
.51614 
-51651 

1.94162 
1-94023 
1-93885 
1.93746 
1-93608 

45 
44 
43 
42 
41 

30 

21 
22 
23 
24 

.44359 
.44385 
.44411 
•44437 
.44464 

.89623 
.89610 
.89597 
-89584 
.89571 

.49495 
.49532 
.49568 
.49604 
.49640 

2-02039 
2-01891 
2-01743 
2-01596 
2-01449 

.45917 
.45942 
.45968 
45994 
.46020 

.88835 

-88822 
88808 
-88795 
-88782 

-51688 
•51724 
•51761 
.51798 
.51835 

1-93470 
1-93332 
1.93195 
1-93057 
1-9?920 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

.44490 
.44516 
.44542 
.44568 
.44594 

.89558 
.89545 
.89532 
.89519 
.89506 

-49677 
.49713 
.49749 
.49786 
-49822 

2-01302 
2-01155 
2-01008 
2-00862 
2-00715 

.46046 
.46072 
•46097 
.46123 
-46149 

-88768 
-88755 
-88741 
.88728 
.88715 

.51872 
•51909 
•51946 
.51983 
-52020 

1-92782 
1-92645 
1-92508 
1-92371 
1-92235 

35 
34 
33 
32 
31 

30 

81 
32 
33 
34 

.  44620 
.44646 
-44672 
.44698 
.44724 

•89493 
.89480- 
.89467 
.89454 
.89441 

.49858 
.49894 
.49931 
.49967 
.50004 

2-00569 
2-00423 
2-00277 
2-00131 
1.99986 

.46175 
.46201 
.46226 
.46252 
•46278 

.88701 
-88688 
-88674 
-88661 
.88647 

.52057 
•52094 
.52131 
.52168 
.52205 

1-92098 
1-91962 
1-91826 
1-91690 
1.91554 

30 

29 
28 
27 
28 

35 
36 
37 
38 
39 

•44750 
.44776 
.44802 
.44828 
.44854 

.89428 

.89415 
.89402 
.89389 
.89376 

.  50040 
.50076 
.50113 
.50149 
.50185 

1-99841 
1-99695 
1-99550 
1-99406 
1-99261 

.46304 
.46330 
•46355 
•46381 
-46407 

•88634 
-88620 
-88607 
-88593 
-88580 

-52242 
.52279 
-52316 
.52353 
-52390 

1-91418 
1-91282 
1.91147 
1.91012 
1.90876 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

.44880 
.44906 
.44932 
44958 
.44984 

.89363 
.89350 
.89337 
.89324 
.89311 

.50222 
.50258 
.50295 
.50331 
.50368 

1-99116 
1-98972 
1-98828 
1-98684 
1.98540 

•46433 
-46458 
-46484 
-46510 
-46536 

-88566 
-88553 
-88539 
-88526 
-88512 

-52427 
.52464 
.52501 
-52538 
.52575 

1-90741 
1-90607 
1-90472 
1-90337 
1-90203 

3O 

19 
18 
17 
16 

45 
46 
47 
48 
49 

.45010 
•45036 
.45062 
.45088 
-45114 

.89298 
.89285 
.89272 
•89259 
.89245 

.  50404 
.  50441 
.50477 
.50514 
.50550 

1-98396 
1-98253 
1.98110 
1.97966 
1-97823 

-46561 
-46587 
.46613 
-46639 
.46664 

-88499 
-88485 
-88472 
-88458 
.  88445 

.52613 
•52650 
-52687 
.52724 
.92761 

1-90069 
1-89935 
1-89801 
1-89667 
1-89533 

15 

14 

13 
12 
11 

50 

51 
52 
53 
54 

-45140 
.45166 
45192 
.45218 
.45243 

.89232 
.89219 
-89206 
.89193 
-89180 

.50587 
-50623 
50660 
-50696 
-50733 

1-97681 
1-97538 
1-97395 
1-97253 
1-97111 

.46690 
.46716 
.46742 
.46767 
.46793 

.88431 
88417 
88404 
88390 
88377 

•52798 
.52836 
.52873 
.52910 
.52947 

1-89400 
1-89266 
1.89133 
1.89000 
1-88867 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

.45269 
.45295 
.45321 
-45347 
•45373 

-89167 
-89153 
-89140 
.89127 
-89114 

-50769 
-50806 
-50843 
-50879 
50916 

1-96969 
1-96827 
1.96685 
1-96544 
1-96402 

-46819 
-  46844 
-46870 
-46896 
-46921 

88363 
88349 
88336 
88322 
88308 

52985 
53022 
53059 
53096 
53134 

1-88734 
1-88602 
1-88469 
1-88337 
1-88205 

5 

4 
3 

2 

1 

60 

•45399 

.89101 

-50953 

1.96261 

-46947 

88295 

53171 

1-88073 

0 

/ 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

63° 


63° 


254    NATURAL  SINES,   COSINES,   TANGENTS,   COTAN. 

28°  29° 


I 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

\ 

y 

2 
3 

.46947 
.46973 
.46999 
.47024 
.47050 

.88295 
.88281 
.88267 
.88254 
.88240 

-53171 
.53208 
53246 
.53283 
.53320 

1.88073 
1-87941 
1-87809 
1.87677 
1.87546 

48481 
.48506 
-48532 
-48557 
-48583 

87462 
.87448 
•87434 
87420 
.87406 

-55431 
-55469 
.55507 
•55545 
.55583 

i  80405 
1.80281 
1.80158 
1-80034 
1.79911 

60 

59 
58 
57 
56 

5 
6 
7 

9 

.47076 
•47101 
.47127 
.47153 
.47178 

.88226 
88213 
.88199 
.88185 
.88172 

.53358 
.53395 
.53432 
53470 
.53507 

1.87415 
.1.87283 
1-87152 
1.87021 
1-86891 

-48608 
•48634 
.48659 
48684 
•48710 

.87391 
.87377 
.87363 
•87349 
.87335 

.55621 
-55659 
.55697 
-55736 
-55774 

1  79788 
1  79665 
1.79542 
1.79419 
1-79296 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.47204 
.47229 
.47255 
.47281 
.47306 

.88158 
.88144 
.88130 
.88117 
.88103 

.53545 
.53582 
.53620 
.53657 
.53694 

1.86760 
1.86630 
1.86499 
1.86369 
1.86239 

•48735 
.48761 
48786 
.48811 
•48837 

•87321 
.87306 
.87292 
•87278 
.87264 

-55812 
-55850 
-55888 
.55926 
-55964 

1-79174 
1  7905] 
1-78929 
1-78807 
1-78685 

50 

49 
48 
47 
46 

15 
16 
17 
18 
*9 

•47332 
.47358 
.47383 
.47409 
.47434 

.88089 
.88075 
.88062 
•88048 
.88034 

•53732 
.53769 
-53807 
-53844 
.53882 

1-86109 
1.85979 
.85850 
:  -85720 
-85591 

.48862 
.48888 
.48913 
.48938 
-48964, 

•87250 
.87235 
•87221 
.87207 
-87193 

-56003 
-56041 
-56079 
-56117 
.56156 

1-78563 
1-78441 
1-78319 
1-78198 
1-78077 

45 
44 
43 
42 
,,.41 

20 

21 
22 
23 
24 

•47460 
.47486 
.47511 
-47537 
.47562 

.88020 
.88006 
.87993 
.87979 
.87965 

.53920 
.53957 
.53995 
.54032 
.54070 

.85462 
.85333 
•83204 
1-85075 
1.84946 

.48989 
.49014 
.49040 
-49065 
-49090 

.87178 
.87164 
.87150 
.87136 
.87121 

.56194 
-56232 
-56270 
-56309 
•56347 

1.77955 
1-77834 
1-77713 
1-77592 
1-77471 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

•47588 
•47614 
47639 
.47665 
.47690 

.87951 
.87937 
•87923 
.87909 
.87896 

.54107 
.54145 
•54183 
.  54220 
.  54258 

1.84818 
1-84689 
1-84561 
1.84433 
1-84305 

-49116 
-49141 
-49166 
-49192 
-49217 

•87107 

87093 
.87079 
-87064 
-87050 

-56385 
-56424 
-56462 
.56501 
.56539 

1-77351 
1-77230 
1  77110 
1-76990 
1-76869 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.47716 
.47741 
.47767 
-47793 
.47818 

.87882 
.87868 
.87854 
.87840 
.87826 

•54296 
•54333 
.54371 
.  54409 
-  54446 

1.84177 
1  -  84049 
1.83922 
1.83794 
1.83667 

-49242 
-49268 
-49293 
-49318 
.49344 

-87036 
-87021 
-87007 
-86993 
-86978 

-56577 
-56616 
•  5'6654 
.56693 
.56731 

1.76749 
1-76629 
1.76510 
1-76390 
1.76271 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

.47844 
•47869 
.47895 
.47920 

.47946 

.87812 
.87798 
.87784 
.87770 
-87756 

-  54484 
-54522 
-54560 
-54597 
-54635 

.83540 
-83413 
-83286 
.83159 
-83033 

.49369 
-49394 
-49419 
•49445 
.49470 

-86964 
-86949 
-86935 
-86921 
.86906 

•56769 
.56808 
.56846 
-56885 
-56923 

1-76151 
1-76032 
1-75913 
1-75794 
1-75675 

25 
24 
23 
22 
-21 

40 

41 
42 
43 
44 

.47971 
-47997 
.48022 
.48048 
-48073 

•87743 
.87729 
•87715 
•87701 
.87687 

.54673 
•54711 
-  54748 
-54786 
-54824 

.82906 
.82780 
1-82654 
1-82528 
1-82402 

•49495 
•49521 
•49546 
•49571 
.49596 

-86892 
-86878 
•86863 
•86849 
.86834 

-56962 
•57000 
-57039 
•57078 
.57116 

1-75556 
1  75437 
1-75319 
1-75200 
1.75082 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

.48099 
•48124 
.48150 
•48175 
.48201 

.87673 
.87659 
.87645 
.87631 
.87617 

.54862 
.  54900 
.54938 
.54975 
.55013 

1.82276 
1-82150 
1-82025 
1-81899 
1-81774 

.49622 
-49647 
•49672 
-49697 
-49723 

-86820 
.86805 
.86791 
-86777 
-86762 

.57155 
.57193 
.57232 
.57271 
-57309 

1-74964 
1  .  74846 
1-74728 
1-74610 
1  .  74492 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

.48226 
.48252 
.48277 
.48303 
.48328 

•87603 
.87589 
-87575 
.87561 
•87546 

.55051 
.55089 
.55127 
.55165 
.  55203 

1-81649 
1.81524 
1.81399 
1-81274 
1-81150 

•49748 
.49773 
-49798 
-49824 
-49849 

.86748 
-86733 
-86719 
.86704 
-86690 

.57348 
-57386 
-57425 
.57464 
-57503 

1-74375 
1  .  .  4257 
1  -  74140 
1  -  74022 
1-73905 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

.48354 
-48379 
•48405 
.48430 
.48456 

•87532 
.87518 
.87504 
•87490 
.87476 

•55241 
.55279 
.55317 
•55355 
.55393 

1.81025 
1-80901 
1.80777 
1.80653 
1-80529 

-49874 
49899 
-49924 
49950 
49975 

-86675 
.86661 
.86646 
.86632 
.86617 

•57541 
.57580 
.57619 
.57657 
-57696 

1-73788 
1-73671 
1.73555 
1-73438 
1.73321 

5 

4 
3 
2 
1 

90 

-48481 

.87462 

•55431 

1.80405 

50000 

.86603 

57735 

1-73205 

0 

Cos. 

Sin. 

Cot. 

Tan. 

Cos, 

Sin. 

Cot. 

Tan. 

' 

61° 


60° 


NATURAL  SINES,   COSINES,   TANGENTS,   COTAN.     255 

30°  31° 


/ 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

.Cot,. 

' 

o 

1 

2 
3 

4 

.50000 
.50025 
.50050 
.50076 
.50101 

.86603 
.86588 
.86573 
.88559 
.86544 

-57735 
.57774 
.57813 
.57851 
.57890 

1.73205 
1-73089 
1.72973 
1.72857 
1.72741 

.51504 
•51529 
.51554 
.51579 
.51604 

.85717 
•85702 
•85687 
.85672 
.85657 

•60086 
•60126 
•60165 
•60205 
•60245 

1.66428 
1-66318 
1-66209 
1-66099 
i-  65990 

60 

59 
58 
57 
56 

5 
6 
7 
8 
9 

.50126 

.50151 
.50176 
.50201 
.50227 

.86530 
.86515 
.86501 
.86486 
.86471 

.57929 
.57968 
.58007 
.58046 
.58085 

1-72625 
1.72509 
1.72393 
1-72278 
1.72163 

.51628 
•51653 
•51678 
.51703 
.51728 

•85642 
•85627 
•85612 
.85597 
-85582 

•60284 

.60324 
•60364 
•60403 
.  60443 

1-65881 
1-65772 
1-65663 
1  65554 
1-65445 

55 
54 
53 
52 
51 

10 

11 
12 
13 

14 

.50252 
.50277 
.50302 
.50327 
-50352 

•86457 
.86442 
.86427 
.86413 
.86398 

.58124 
.58162 
.58201 
.58240 
-58279 

1-72047 
1.71932 
1.71817 
1.71702 
1.71588 

-51753 
•51778 
-51803 
•51828 
•51852 

•85567 
•85551 
.85536 
-85521 
.85506 

•60483 
•60522 
.60562 
•60602 
•60642 

1-65337 
1-65228 
1-65120 
1.65011 
1  .  64903 

5O 

49 
48 
47 
48 

15 
16 
17 
18 
19 

.50377 
•50403 
•50428 
.50453 
.50478 

•86384 
.86369 
•86354 
.86340 
.86325 

.58318 
.58357 
.58396 
-58435 
.58474 

1-71473 
1.71358 
1.71244 
1-71129 
1-71015 

.51877 

.51902 
-51927 
•51952 
.51977 

.85491 
.85476 
-85461 
-85446 
85431 

•60681 
.60721 
.60761 
.60801 
.60841 

1-64795 
1-64687 
1-64579 
1-64471 
1-64363 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

.50503 
.50528 
-50553 
.50578 
.50603 

.86310 

•86295 
.86281 
.86266 
.86251 

•58513 
-58552 
-58591 
-58631 
.58670 

1-70901 
1-70787 
1-70673 
1-70560 
1  .  70446 

•52002 
.52026 
•52051 
.52076 
-52101 

-.85416 
.85401 
-85385 
-85370 
-85355 

•60881 
.60921 
•60960 
•61000 
61040 

1-64256 
1-64148 
1  -64041 
1-63934 
1-63826 

40 

39 
38 
37 
36 

25 
28 
27 
28 
89 

.50628 
.50654 
.50679 
-50704 
•50729 

.86237 
•86222 
,86207 
.86192 
.86178 

58709 
.58748 
.58787 
.58826 
.58865 

1-70332 
1-70219 
1-70106 
1-69992 
1.69879 

.52126 
.52151 
.52175 
.52200 
-52225 

•85340 
.85325 
•85310 
.85294 
.85279 

.61080 
.61120 
.61160 
.61200 
-61240 

1-63719 
1-63612 
1-63505 
1-63398 
1-63292 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

•  50754 
.50779 
.  50804 
.50829 
.50854 

.86163 
•86148 
.86133 
•86119 
.86104 

.58905 
.58944 
.58983 
.59022 
.59061 

1-69766 
1-69653 
1-69541 
1-69428 
1-69316 

•52250 
•52275 
.52299 
.52324 
•  52349 

•85264 
.85249 
•85234 
•85218 
.85203 

-61280 
-61320 
.61360 
.61400 
.61440 

1-63185 
1-63079 
1-62972 
1-62866 
1-62760 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

•50879 
.50904 
•50929 
•50954 
•50979 

.86089 
-86074 
.86059 
•86045 
•86030 

.59101 

•59140 
.59179 
59218 
.59258 

1-69203 
1.69091 
1-68979 
1.68866 
1-68754 

.52374 
.52399 
•52423 
•52448 
•52473 

.85188 
•85173 
•85157 
.85142 
-85127 

•61480 
•61520 
•61561 
•61601 

.61641 

1-62654 
1-62548 
1-62442 
1-62336 
1-62230 

25 

24 
23 
22 
21 

40 

41 
42 
43 
44 

•51004 
•51029 
•51054 
.51079 
•51104 

.86015 
.86000 
85985 
-85970 
.85956 

.59297 
.59336 
.59376 
.59415 
.59454 

1-68643 
1.68531 
1.68419 
1-68308 
1.68196 

.52498 
•52522 
•52547 
•52572 
.52597 

.85112 
.85096 
•85081 
•85066 
-85051 

•61681 
•61721 
•61761 
.61801 
.61842 

1-62125 
]-  62019 
1-61914 
1-61808 
1-61703 

20 

19 
18 
17 
18 

45 
46 
47 
48 
49 

•51129 
•  51154 
•51179 
•51204 
.51229 

•85941 
.85926 
.85911 
.85896 
.85881 

.59494 
•59533 
•59573 
.59612 
.59651 

1-68085 
1-67974 
1-67863 
1.67752 
1-67641 

-52621 
-52646 
-52671 
-52696 
.52720 

•85035 
•85020 
•85005 
-84989 
-84974 

•61882 
.61922 
61962 
.62003 
•62043 

1-61598 
1.61493 
1.61388 
1-61283 
1-61179 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

.51254 
•51279 
•51304 
•51329 
.51354 

•85866 
.85851 
85836 
•85821 
.85806 

•59691 
•59730 
•59770 
•59809 
•59849 

1-67530 
1-67419 
1-67309 
1  67198 
1-67088 

.52745 
-52770 
-52794 
-52319 
.52844 

•84959 
.  84943 
.84928 
•84913 
.84897 

•62083 
•62124 
•62164 
•62204 
.62245 

1-61074 
1-60970 
1  60865 
1-60761 
1.60657 

10 
9 
8 
7 
6 

55 
56 
57 
58 
59 

•51379 
51404 
.51429 
•51454 
•51479 

•85792 
•85777 
.85762 
.85747 
85732 

.59888 
•59928 
.59967 
•60007 
•  60046 

1-66978 
1-66867 
1-66757 
1-66647 
1-66538 

.52869 
.52893 
-52918 
.52943 
.52967 

•84882 
•84866 
.84851 
•84836 
.84820 

•62285 
•62325 
•62366 
•62406 
.  62446 

1.60553 
1  .  60449 
1-60345 
1  •  60241 
1-60137 

5 
4 
3 
2 
1 

(60 

.51504 

•85717 

•60086 

1-66428 

•52992 

.84805 

•62487 

1  -  60033 

O 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

69° 


58° 


256    NATURAL  SINES,   COSINES,   TANGENTS,   COTAN. 

33°  33° 


t 

Sin. 

Cos 

Tan. 

Cot 

Sin. 

Cos. 

Tan. 

Cot. 

* 

o. 

2 
3 

.52992 
.53017 
.53041 
.53066 
.53091 

.84805 
•84789 
.84774 
.84759 
.  84743 

.62487 
•62527 
.62568 
.62608 
•62649 

-60033 
.59930 
.59826 
•59723 
.59620 

•54464 
-54488 
•54513 
•54537 
-54561 

.83867 
.83851 
•83835 
•83819 
-83804 

•64941 
•64982 
•65024 
•65065 
•65106 

1-53986 
1-53888 
1-53791 
1-53693 
1-53595 

60 

59 
58 
57 
56 

6 

j 

9 

.53115 
.53140 
.53164 
.53189 
.53214 

.84728 
•84712 
•84697 
•84681 
.84666 

-62689 
.62730 
.62770 
.62811 
-62852 

•59517 
.59414 
.59311 
.59208 
.59105 

-  54583 
.54610 
•54635 
•54659 
.  54683 

-83788 
•83772 
•83756 
•83740 
.83724 

•65148 
•65189 
-65231 
•65272 
•65314 

1-53497 
1-53400 
1-53302 
1-53205 
1.53107 

55 
54 
53 
5L 
51 

10 

i* 

13 
14 

.53238 
.53263 
.53288 
.53312 
.53337 

•84650 
.84635 
•84619 
.84604 
.84588 

.62892 
.62933 
-62973 
-63014 
.63055 

.59002 
.58900 
.58797 
.58695 
.58593 

-54708 
-54732 
-54756 
-54781 
-54805 

•83708 

•83692 
•83676 
•83660 
•83645 

•65355 
-65397 
-65438 
-65480 
-65521 

1-53010 
1-52913 
1-52816 
1-52719 
1-52622 

50 

49 
48 
47 
48 

15 
16 
17 
18 
19.. 

.53361 
.53386 
.53411 
.53435 
.53460 

.84573 
.84557 
.84542 
.84526 
.84511 

.63095 
-63136 
.63177 
.63217 
-63258 

.58490 
.58388 
-68286 
.58184 
-58083 

-54829 
.54854 
.54878 
•54902 

-54927 

•83629 
•83613 
•83597 
•83581 
.83565 

•65563 

-65604 
•65646 
•65688 
-65729 

1-52525 
1-52429 
1-52332 
1-52235 
1-52139 

45 
44 
43 
42 
41 

2O 

21 
22 
23 
24 

-53484 
.53509 
.53534 
.53558 
.53583 

.84495 
•84480 
.84464 
.  84448 
.84433 

-63299 
-63340 
-  63380 
.63421 
.  63462 

.57981 
'  .57879 
.57778 
.57676 
.57575 

54951 
-54975 
•54999 
•55024 
.55048 

.83549 
.83533 
•83517 
•83501 
•83485 

.65771 
.65813 
•65854 
.65896 
.65938 

1.52043 
1.51946 
1-51850 
1-51754 
1.51658 

4O 

39 
38 
37 
36 

25 
26 
27 
28 
29 

•53607 

.53632 
.53656 
-53681 
.53705 

•84417 
.  84402 
•84386 
.84370 
.84355 

•63503 
.63544 
.63584 
.63625 
.63666 

.57474 
.57372 
.57271 
.57170 
.57069 

•55072 
-55097 
•55121 
-55145 
.55169 

•83469 
•83453 
•83437 
•83421 
•83405 

.65980 

.66021 
.66063 
.66105 
-66147 

1-51562 
1-51466 
1-51370 
1-51275 
1.51179 

35 
34 
33 
32 
31 

30 

81 
32 
33 
34 

.53730 
.53754 
.53779 
.53804 
.53828 

.84339 

.84324 
.84308 
.84292 
.84277 

.63707 
-63748 
-63789 
.63830 
.63871 

.56969 
.56868 
.56767 
.56667 
.56566 

.55194 
.55218 
.55242 
.55266 
-55291 

•83389 
•83373 
-83356 
-83340 
-83324 

.66189 

.66230 
.66272 
.66314 
.66356 

1-51084 
1.50988 
1-50893 
1.50797 
1.50702 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

.53853 
.53877 
.53902 
.53926 
.53951 

•84261 

.84245 
-84230 
.84214 
.84198 

.63912 
.63953 
-63994 
.64035 
-64076 

.56466 
.56366 
•56265 
.56165 
1-56065 

.55315 
-55339 
.55363 
.55388 
-55412 

•83308 
•83292 
•83276 
•83260 
•83244 

.66398 
.66440 
•66482 
.66524 
.66566 

1-50607 
1-50512 
1.50417 
1.50322 
1.50228 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

.53975 
.  54000 
.54024 
.54049 
.54073 

-84182 
.84167 
.84151 
.84135 
.84120 

.64117 
.64158 
.64199 
.  64240 
-64281 

1.55966 
1.55866 
1.55766 
1.55666 
1-55567 

-55436 
.55460 
.55484 
-55509 
-55533 

•83228 
.83212 
33195 
.83179 
•83163 

.66608 
.66650 
.66692 
.66734 
.66776 

1.50133 
1.50038 
1-49944 
1.49849 
1-49755 

30 

19 
18 
17 
16 

45 
46 
47 
48 
49 

.54097 
.54122 
.54146 
.54171 
.54195 

•84104 
.84038 
-84072 
-84057 
.  84041 

.64322 
.64363 
.  64404 
.  64446 
-  64487 

1-55467 
1.55368 
1.55269 
1.55170 
1.55071 

.55557 
-55581 
-55605 
.55630 
55654 

.83147 
.83131 
.83115 
•83098 
-83082 

.66818 
.66860 
.66902 
.66944 
.66986 

1.49661 
1-49566 
1.49472 
1.49378 
1.49284 

15 
14 
13 
12 
11 

50 

51 
59, 
53 
54 

.54220 
.  54244 
.54269 
.54293 
.54317 

•84025 
-84009 
.83994 
-83978 
•83962 

-64528 
•64569 
-64610 
•64652 
-64693 

1.54972 
1.54873 
1  .  54774 
1.54675 
1.54576 

.55678 
-55702 
•55726 
-55750 
.55775 

•83066 
-83050 
.83034 
.83017 
83001 

.67028 
.67071 
.67113 
.67155 
.67197 

1.49190 
1-49097 
1.49003 
1.48909 
1.48816 

1O 

9 
8 
7 
6 

55 
56 
57 
58 
59 

•54342 
•54366 
•54391 
.54415 
.  54440 

•83946 
•83930 
.83915 
-83899 
.83883 

.  64734 
-64775 
.64817 
-64858 
64899 

1  .  54478 
1-54379 
1.54281 
1.54183 
1-54085 

•55799 
-55823 
-55847 
.55871 
-55895 

82985 
82969 
-82953 
•82936 
•82920 

.67239 
.67282 
.67324 
.67366 
.67409 

1.48722 
1.48629 
1.48536 
1-48442 
1-48349 

5 

3 
2 

60 

.  54464 

.83867 

-64941 

1-53986 

.55919 

82904 

.67451 

1-48256 

.O 

' 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

66° 


NATURAL  SINES,   COSINES,   TANGENTS,   COTAN.     257 

34°  35° 


Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

/ 

o 

2 

4 

.55919 
•  55943 
.55968 
.55992 
.56016 

-82904 
•82887 
•82871 
•82855 
82839 

•67451 
•67493 
.67536 
•67578 
•67620 

1-48256 
1.48163 
1.48070 
1-47977 
1.47885 

•57358 
•57381 
•57405 
•57429 
•57453 

-81915 
•81899 
•81882 
•81865 
-81848 

-70021 
-70064 
-70107 
.70151 
.70194 

1.42815 
1-42726 
1-42638 
1-42550 
1-42462 

60 

59 
58 
57 
56 

5 

? 
1 

-56040 
.56064 
.56088 
.56112 
.56136 

.82822 
•82806 
•82790 
.82773 
.82757 

•67663 
•67705 
•67748 
•67790 
•67832 

1-47792 
1-47699 
1-47607 
1-47514 
1-47422 

-57477 
-57501 
-57524 
-57548 
-57572 

-81832 
-81815 
•81798 
-81782 
-81765 

•70238 
•70281 
.70325 
•70368 
•70412 

1-42374 
1-42286 
1-42198 
1-42110 
1-42022 

55 
54 
53 
52 
51 

1O 
11 
12 
13 
14 

•  56160 
.56184 
.56208 
.56232 
•56256 

.82741 
.82724 
.82708 
.82692 
.82675 

•67875 
•67917 
.67960 
.68002 
.68045 

1-47330 
1-47238 
-47146 
•47053 
•46962 

-57596 
•57619 
•57643 
•57667 
•57691 

-81748 
•81731 
•81714 
-81698 
-81681 

.  70455 
.70499 
•70542 
•70586 
•70629 

1-41934 
1-41847 
1-41759 
1-41672 
1-41584 

5O 

49 
48 
47 
46 

15 
16 
17 
18 
19 

.56280 
.56305 
.56329 
•56353 
•56377 

•82659 
.82643 
.82626 
.82610 
-82593 

.68088 
.68130 
•68173 
-68215 
•68258 

.46870 
.46778 
.46686 
.46595 
.46503 

.57715 
.57738 
.57762 
•57786 
•57810 

•81664 
•81647 
•81631 
•81614 
•81597 

•70673 
•70717 
•70760 
•  70804 
•  70848 

1-41497 
1-41409 
1-41322 
1-41235 
1-41148 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

.56401 
.56425 
.  56449 
.56473 
.56497 

•82577 
•82561 
•82544 
•82528 
•82511 

.68301 
.68343 
•68386 
.68429 
•68471 

•46411 
-46320 
-46229 
.46137 
1-46046 

.57833 
•57857 
•57881 
•57904 
•57928 

-81580 
•81563 
•81546 
•81530 
•81513 

•70891 
•70935 
•70979 
.71023 
-71066 

1-41061 
1-40974 
1-40887 
1-40800 
1-40714 

40 
39 
38 
37 
36 

25 
26 
27 
28 
29 

•56521 
-56545 
.56569 
-56593 
.56617 

•82495 
•82478 
•82462 
.82446 
.82429 

•68514 
.68557 
•68600 
•68642 
•68685 

1-45955 
1.45864 
1-45773 
1-45682 
1-45592 

-57952 
•57976 
•57999 
•58023 
•58047 

-81496 
•81479 
•81462 
-81445 
-81428 

-71110 
-71154 
-71198 
-71242 
-71285 

1.40627 
1  •  40540 
1.40454 
1.40367 
1-40281 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.56641 
.56665 
.56689 
.56713 
-56736 

.82413 
.82396 
.82380 
•82363 
.82347 

•68728 
•68771 
•68814 
.68857 
.68900 

1-45501 
1-45410 
1-45320 
1-45229 
1-45139 

-58070 
•58094 
•58118 
•58141 
-58165 

-81412 
-81395 
-81378 
-81361 
-81344 

.71329 
-71373 
-71417 
-71461 
-71505 

1-40195 
1-40109 
1-40022 
1-39936 
1.39850 

3O 

29 
28 
27 
26 

35 
36 
37 
38 
39 

-56760 
.  5678-ft 
.  56803 
•56832 
.568?i6_ 

-82330- 

.82314 
.82297 
•82281 

-82264 

.68942 
68985 

.69028 
.69071 
.69114 

1.45049 
1.44958 
1-44868 
1-44778 
1-44688 

-58189 
-58212 
-58236 
-58260 
-58283 

-81327 
-81310 
•81293 
•81276 
•81259 

-71549 
•71593 
.71637 
.71681 
.71725 

1-39764 
1-39679 
1-395'S 
1-39507 

1-39421 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

•  56880 
.56904 
.56928 
•56952 
.56976 

.82248 
.82231 
.82214 
•82198 
-82181 

.69157 
-69200 
.69243 
-69286 
-69329 

1-44598 
1-44508 
1-44418 
1-44329 
1-44239 

-58307 
-58330 
-58354 
-58378 
-58401 

•81242 
.81225 
•81208 
•81191 
•81174 

.71769 
.71813 
.71857 
.71901 
•71946 

1-39336 
1-39250 
1.39165 
1-39079 
1-38994 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

.  57000 

.57024 
.57047 
.57071 
.57095 

.82165 
•82148 
.82132 
•82115 
.82098 

-69372 
.69416 
•69459 
.69502 
•69545 

1.44149 
1  •  44060 
1-43970 
1-43881 
1-43792 

•  58425 
.  58449 
•58472 
•58496 
.58519 

-81157 
•81140 
•81123 
81106 
-81089 

-71990 
.72034 
.72078 
.72122 
•72167 

1.38909 
1.38824 
1.38738 
1-38653 
1-38568 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

.57119 
•57143 
.57167 
.57191 
•57215 

•82082 
•82065 
•82048 
•82032 
•82015 

-69588 
•69631 
•69675 
•69718 
•69761 

1-43703 
1-43614 
1-43525 
1-43436 
1-43347 

-58543 
-58567 
-58590 
-58614 
-58637 

•81072 
•81055 
.81038 
-81021 
•81004 

.72211 
•72255 
•72299 
•  72344 
.72388 

1-38484 
1-38399 
1-38314 
1-38229 
1-38145 

10 
9 
8 
7 

6 

55 
56 
57 
58 
59 

•57238 
•  57262 
.57286 
•57310 
•57334 

•81999 

-81982 
-81965 
81949 
81932 

•69804 
•69847 
•69891 
-69934 
•69977 

1-43258 
1-43169 
1-43080 
1-42992 
1-42903 

•58661 
-58684 
-58708 
.58731 
•58755 

•80987 
.80970 
•80953 
•80936 
•80919 

.  72432 
.72477 
•72521 
.72565 
•72610 

1-38060 
1-37976 
1-37891 
1-37807 
1-37722 

5 
4 
3 

2 

60 

•57358 

81915 

.70021 

1-42815 

-58779 

-80902 

.72654 

L37638 

o 

* 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

65° 


64° 


258    NATURAL  SINES,   COSINES,   TANGENTS,   COTAN. 
36°  37° 


Sin. 

Cos. 

Tan. 

Cot. 

Sin.  1 

Cos. 

Tan. 

Cot. 

, 

0 

2 
3 

•58779 
•58802 
-58826 
•58849 
.58873 

.80902 
•80885 
•80867 
•80850 
•80833 

.72654 
•72699 
.72743 
.72788 
•72832 

1.37638 
1-37554 
1-37470 
1-37386 
1-37302 

•60182 
60205 
60228 
60251 
60274 

-79864 
-  79846 
-79829 
.79811 
•79793 

•75355 
•75401 
•  75447 
-75492 
-75538 

1-32704 
1-32624 
1-32544 
1-32464 
1-32384 

60 

59 
58 
57 

56 

5 
6 
7 
8 
9 

58896 
58920 
58943 
58967 
58990 

•80816 
•80799 
•80782 
•80765 
•80748 

•72877 
•72921 
•72966 
•73010 
•73055 

1-37218 
1.37134 
1.37050 
1-36967 
1.36883 

60298 
60321 
•60344 
-60367 
-60390 

•79776 
-79758 
.79741 
.79723 
-79706 

•75584 

•75629 
•75675 
-75721 
-75767 

1-32304 
1-32224 
1-32144 
1-32064 
1-31984 

55 
54 
53 
52 
51 

10 

11 
12 

13 
14 

59014 
59037 
59061 
59084 
59108 

•80730 
•80713 
•80696 
•80679 
•80662 

73100 

73144 
73189 
73234 
73278 

1-36800 
1.36716 
1.36633 
1.36549 
1.36466 

•  60414 
•60437 
60460 
•60483 
.60506 

-79688 
•79671 
-79653 
-79635 
-79618 

•75812 
-75858 
-75904 
•75950 
-75996 

1-31904 
1-31825 
1-31745 
1-31666 
1-31586 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

59131 
59154 
59178 
59201 
59225 

•80644 
•80627 
80610 
80593 
80576 

73323 
73368 
73413 
73457 
73502 

1-36383 
1.36300 
1.36217 
1.36134 
1-36051 

60529 
60553 
60576 
60599 

60622 

•79600 
•79583 
-79565 
-79547 
-79530 

•76042 
•76088 
•76134 
•76180 
76226 

1-31507 
1-31427 
1-31348 
1-31269 
1-31190 

45 
44 
43 
42 
41 

«0 

21 
22 
23 
24 

59248 
59272 
59295 
59318 
59342 

80558 
80541 
80524 
80507 
80489 

73547 
73592 
73637 
73681 
73726 

1.35968 
1-35885 
1-35802 
1-35719 
1-35837 

60645 
60668 
60691 
•  '$0714 
•60738 

79512 
.79494 
.79477 
•79459 
-  79441 

•76272 
•76318 
•76364 
•76410 
•76456 

1-31110 
1.31031 
1.30952 
1-30873 
1  30795 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

.59385 
59389 
59412 
59436 
59459 

80472 
80455 
80438 
80420 
80403 

73771 
73816 
73861 
73906 
73951 

1-35554 
1-35472 
1-35389 
1.35307 
1-35224 

•60761 
•6C784 
•60807 
•60830 
•60853 

-  79424 
•79406 
.79388 
•79371 
•79353 

-76502 
-76548 
-76594 
-76640 
.76686 

1-30716 
1-30637 
1-30558 
1  -  30480 
1  •  30401 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

59482 
59506 
59529 
59552 
59576 

80386 
•80368 
80351 
80334 
•80318 

73996 
74041 
74086 
74131 
74176 

1.35142 
1-35060 
1.34978 
1.34896 
1-34814 

•60876 
•60899 
60922 
•60945 
-60968 

.79335 
-79318 
-79300 
-79282 
-79264 

.76733 
.78779 
•76825 
•76871 
•76918 

1-30323 
1-30244 
1-30166 
1-30087 
1-30009 

30 

29 
28 
27 

26 

35 
36 
37 
38 
39. 

59599 
59622 
59646 
59669 

59693. 

•80299 
-80282 
-80264 
•80247 
•80230 

74221 
74267 
74312 
74357 
74402 

1-34732 
1.34650 
1-34568 
1.34487 

1.34405 

-60991 
-61015 
-61038 
•61061 

•61084 

79247 
.79229 
.79211 
-79193 
-79176 

•76964 
.77010 
-77057 
.77103 
.77149 

1-29931 
1.29853 
1-29775 
1-29696 
1-29B18 

25 
24 
23 
22 
21 

40 

41 

42 
43 
44 

.59716 
.59739 
.59763 
.59786 
.59809 

.80212 
•80195 
•80178 
•80160 
•80143 

74447 
74492 
•74538 
•74583 
.74628 

1.34323 
1-34242 
1-34160 
1-34079 
1-33998 

•61107 
•61130 
•61153 
•61176 
•61199 

-79158 
-79140 
-79122 
-79105 
-79087 

.77196 
•77242 
-77289 
-77335 
-77382 

1-29541 
1-29463 
1-29385 
1-29307 
1-29229 

20 

19 
18 
17 
Ifi 

45 
46 
47 
48 
49_ 

59832 
.59856 
.59879 
•59902 
.59926 

.80125 
.80108 
.80091 
.80073 
.80056 

•74674 
.74719 
•74764 
•74810 
.74855 

1-33916 
1-33835 
1.33754 
1.33673 
1-33592 

-61222 
•61245 
-61268 
-61291 
-61314 

-79069 
-79051 
-79033 
-79016 
-78998 

-77428 
•77475 
.77521 
.77568 
.77615 

1-29152 
1-29074 
1-28997 
1-28919 
1-28842 

15 
14 
13 
12 
11 

50 

51 
52 
53 
5i_ 

•59949 
•59972 
.59995 
•60019 
.60042 

.80038 
.80021 
.80003 
.79986 
.79968 

•74900 
.74946 
•74991 
•75037 
.75082 

1-33511 
1.33430 
1.33349 
1.33268 
1-33187 

-61337 
-61360 
.61383 
•  6H06 
•61429 

-78980 
-78962 
-78944 
-78926 
-78908 

.77661 
.77708 
•77754 
-77801 
-77848 

1-28764 
1-28687 
1-28610 
1-28533 
1-28456 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

•60065 
•60089 
•60112 
•60135 
•60158 

.79951 
.79934 
.79916 
.79899 
.79881 

.75128 
•75173 
.75219 
•75264 
.75310 

1.33107 

1.33026 
1.32946 
1-32865 
1-32785 

.61451 
•61474 
•61497 
•61520 
-61543 

-78891 
-78873 
-78855 
•78837 
78819 

.77895 
-77941 
•77988 
•78035 
.78082 

1-28379 
1-28302 
1-28225 
1-28148 
1-28071 

5 
4 
3 
2 

1 

60 

.60182 

.79864 

,75355 

1-32704 

-61566 

•78801 

.78129 

1-27994 

0 

. 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

63* 


NATURAL  SINES,   COSINES,   TANGENTS,   COTAN.    259 
38°  39° 


1 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

' 

o 

1 

2 
3 
4 

•61566 
-61589 
.61612 
-61635 
-61658 

-78801 
•78783 
-78765 
•78747 
-78729 

•78129 
•78175 
•78222 
•78269 
-78316 

1-27994 
1-27917 
1-27841 
1-27764 
1-27688 

•62932 
.62955 
-62977 
-  63000 
-63022 

•77715 
•77696 
•77678 
•77660 
•77641 

•80978 
•81027 
•81075 
•81123 
-81171 

1-23490 
1-23416 
1-23343 
1-23270 
1-23196 

60 

59 
58 
57 
56 

5 
6 
7 
8 

9 

-61681 
-61704 
-61726 
-61749 
.61772 

•78711 
-78694 
-78676 
-78658 
-  78640 

•78363 
•78410 
•  78457 
-78504 
-78551 

•27611 
.27535 
•27458 
•27382 
•27306 

•  63045 
•63068 
-63090 
.63113 
-63135 

•77623 
•77605 
77586 
•77568 
.77550 

•81220 
•81268 
•81316 
•81364 
•81413 

1-23123 
1-23050 
1-22977 
1-22904 
1-22831 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

-61795 
-61818 
-61841 
-61864 
-61887 

-78622 
•78604 
•78586 
•78568 
-78550 

-78598 
-78645 
-78692 
•78739 
•78786 

•27230 
•27153 
.27077 
•27001 
•26925 

-63158 
-63180 
-63203 
-63225 
.63248 

.77531 
.77513 
.77494 
.77476 
.77458 

.81461 
.81510 
-81558 
-81606 
.81655 

1-22758 
1-22685 
1.22612 
1-22539 
1-22467 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

-61909 
-61932 
-61955 
-61978 
.62001 

.78532 
•78514 
-78496 
-  78478 
.78460 

-78834 
-78881 
•78928 
-78975 

.79022 

.26849 
•26774 
.26698 
.26622 
-26546 

•63271 
•63293 
•63316 
-63338 
.63361 

.  77439 
•  77421 
.  77402 
•77384 

.77366 

.81703 
.81752 
-81800 
-81849 
.81898 

1-22394 
1-22321 
1-22249 
1.22-176 
1-22104 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

•62024 
•62046 
.62069 
-62092 
•62115 

-78442 
-78424 
.78405 
-78387 
•78369 

•79070 
•79117 
-79164 
•79212 
-79259 

.26471 
1-26395 
1-26319 
1-26244 
1-26169 

-63383 
-63406 
.63428 
.63451 
.63473 

•77347 
•77329 
•77310 
.77292 
.  77273 

.81946 
•81995 
-82044 
-82092 
•82141 

1.22031 
1-21959 
1-21886 
1-21814 
1-21742 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

-62138 
-62160 
.82183 
.  62206 
•62229 

•78351 
.78333 
-78315 
-78297 
•78279 

•79306 
•79354 
•79401 
•  79449 
•79496 

1-26093 
1.26018 
1-25943 
1.25867 
1-25792 

-63496 
.63518 
-63540 
-63563 
-63585 

•77255 
•77236 
•77218 
•77199 
•77181 

•82190 
•82238 
•82287 
-82336 
-82385 

1.21670 
1.21598 
1.21526 
1-21454 
1-21382 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

-62251 
•62274 
•62297 
•62320 
-62342 

•78261 
-  78243 
-78225 
-78206 
-78188 

•79544 
•79591 
•79639 
•79686 
•79734 

1-25717 
1-25642 
1-25567 
1-25492 
1.25417 

-63608 
-63630 
-63653 
-63675 
•63698 

•77162 
.77144 
.77125 
.77107 
.77088 

•  82434 
•82483 
-82531 
-82580 
-82629 

1-21310 
1.21238 
1-21166 
1-21094 
1-21023 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

-62365 
-62388 
-62411 
.62433 
•62456 

•78170 
•78152 
-78134 
•78116 

.78098 

-79781 
.79829 
.79877 
•79924 
-79972 

1-25343 
1-25268 
1-25193 
1-25118 
1.25044 

-63720 
-63742 
-63765 
-63787 
-63810 

.77070 
•77051 
.77033 
•77014 
-76996 

•82678 
-82727 
-82776 
•82825 
•82874 

1-20951 
1-20879 
1-20808 
1-20736 
1-20665 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

•62479 
.62502 
-62524 
-62547 
.62570 

.78079 
•78061 
.  78043 
•78025 
•78007 

•80020 
-80067 
.80115 
.80163 
•80211 

1-24969 
1-24895 
1.24820 
1.24746 
1.24672 

-63832 
.63854 
•63877 
.63899 
-63922 

-76977 
-76959 
-76940 
-76921 
•76903 

•82923 
-82972 
•83022 
•83071 
-83120 

1-20593 
1-20522 
1-20451 
1-20379 
1-20308 

20 

19 
18 
17 
16 

45 
46 
47 
48 
49 

•62592 
•62615 
-62638 
-62860 
-62683 

-77988 
•77970 
•77952 
•77934 
•77916 

-80258 
-80306 
-80354 
-80402 
.  80450 

1-24597 
1-24523 
1  •  24449 
1.24375 
1-24301 

•  63944 
•63966 
.63989 
-64011 
-  64033 

•76884 
-76866 
-76847 
-76828 
•76810 

-83169 
-83218 
-83268 
-83317 
-83366 

1-20237 
1-20166 
1-20095 
1  -  20024 
1-19953 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

-62706 
-62728 
-62751 
-62774 
-62796 

•77897 
-77879 
•77861 
•77843 
-77824 

-80498 
-80546 
-80594 
-80642 
-80690 

1-24227 
1-24153 
1  -  24079 
1  -  24005 
1.23931 

-64056 
-64078 
-64100 
-64123 
•64145 

•76791 
•76772 
•76754 
•76735 
•76717 

•83415 
-83465 
-83514 
-83564 
-83613 

1-19882 
1-19811 
1-19740 
1-19669 
1-19599 

10 

9 
8 
7 
6 

55 
56 
57 
58 
59 

•62819 
-62842 
.62864 
.62887 
-62909 

-77806 
-77788 
-77769 
•77751 
-77733 

.80738 
•80786 
.80834 
.80882 
•80930 

1-23858 
1-23784 
1-23710 
1-23637 
1-23563 

-64167 
.64190 
.64212 
.  64234 
-64256 

•76698 
•76679 
•76661 
•76642 
•76623 

-83662 
-83712 
-83761 
-83811 
-83860 

1-19528 
1-19457 
1-19387 
1-19316 
1-19246 

5 

4 
3 

2 

1 

60 

-62932 

-77715 

80978 

1-23490 

•64279 

-76604 

.83910 

1-19175 

0 

' 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

' 

61° 


50° 


260    NATURAL  SINES,   COSINES,   TANGENTS,   COTAN. 

40°  41° 


/ 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

t 

o 

1 

2 
3 
4 

64279 
64301 
64323 
64346 
64368 

76604 
76586 
76567 
76548 
76530 

83910 
83960 
84009 
84059 
•84108 

1-19175 
1-19105 
1-19035 
1-18964 
1-18894 

65606 
65628 
65650 
65672 
65694 

75471 
75452 
75433 
75414 
75395 

86929 
86980 
87031 
87082 
87133 

1-15037 
1-14969 
1.14902 
1.14834 

1-14767 

60 

59 
58 
57 
56 

5 
6 

8 
9 

64390 
64412 
64435 
64457 
64479 

76511 
76492 
76473 
76455 
76436 

•84158 
•84208 
•84258 
•84307 
.84357 

1-18824 
1-18754 
1-18684 
1-18614 
1-18544 

65716 
65738 
65759 
65781 
65803 

75375 
75356 
75337 
•75318 
•75299 

87184 
87236 
87287 
-87338 
-87389 

1-14699 
1.14632 
1-14565 
1-14498 
1.14430 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

64501 

64524 
64546 
64568 
64590 

.76417 
•76398 
.76380 
•76361 
76342 

.  84407 
•84457 
-84507 
-84556 
-84606 

1-18474 
1-18404 
1-18334 
1-18264 
1-18194 

65825 
65847 
65869 
65891 
65913 

•75280 
•75261 
-75241 
-75222 
•75203 

•87441 
•87492 
-87543 
•87595 
•87646 

1-14363 
1-14296 
1-14229 
1-14162 
1-14095 

50 

49 
48 
47 
46 

15 
16 
17 
18 
19 

64612 
64635 
64657 
64679 
64701 

.76323 
.76304 
.76286 
.76267 

•76248 

•84656 
.84706 
•84756 
.84806 
•84856 

1.18125 
1-18055 
1.17986 
1-17916 
1-17846 

-65935 

-  65956 
-65978 
-  66000 
.66022 

•75184 
•75185 
•75146 
-75126 
-75107 

•87698 
•87749 
•87801 
-87852 
.87904 

1-14028 
1-13961 
1-13894 
1-13828 
1-13761 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

64723 
64746 
64768 
64790 
64812 

•76229 
•76210 
•76192 
•76173 
•76154 

.84906 
•  i/4956 
•85006 
•85057 
.85107 

1.17777 
1-17708 
1-17638 
1-17569 
1-17500 

-  66044 
-66066 
-66088 
-66109 
-66131 

-75088 
-75069 
-75050 
-75030 
•75011 

•87955 
.88007 
.88059 
•88110 
•88162 

1-13694 
1-13627 
1-13561 
1-13494 
1-13428 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

.64834 
•64856 
.64878 
•64901 
.64923 

•76135 
•76116 
•76097 
•76078 
•76059 

•85157 
•85207 
•85257 
-85308 
-85358 

1-17430 
1-17361 
1.17292 
1-17223 
1.17154 

-66053 
•66175 
•66197 
•66218 
.  66240 

•  74992 
•74973 
•74953 
•  74934 
•74915 

•88204 
•88265 
•88317 
•88369 
.88421 

1-13361 
1-13295 
1-13228 
1-13162 
1.13096 

35 
34 
33 
32 
31 

30 

31 
32 
33 
34 

.64945 
•64967 
•64989 
•65011 
-65033 

•  76041 
.76022 
•  76003 
•75984 
•75965 

•85408 
•85458 
•85509 
.85559 
•85609 

1-17085 
1-17016 
1-16947 
1-16878 
1-16809 

-66262 
•66284 
•66306 
-66327 
•66349 

•74896 
•74876 
•74857 
•74838 
•74818 

.88473 
.88524 
.88576 
.88628 
-88680 

1-13029 
1-12963 
1-12897 
1-12831 
1-12765 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

•65055 
•65077 
.65100 
.65122 

-65144 

•75946 
•75927 
.75908 
-75889 
-75870 

.85660 
•85710 
•85761 
-85811 
-85862 

1-16741 
1-16672 
1-16603 
1-16535 
1-16466 

-66371 
-66393 
-66414 
•66436 
-66458 

•74799 
.  74780 
.  74760 
.  74741 

-74722 

-88732 
-88784 
-88836 
-88888 
-88940 

1-12699 
1-12633 
1-12567 
1.12501 

1-12435 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

-65166 
.65188 
•65210 
•65232 
-65254 

-75851 
-75832 
-75813 
-75794 
-75775 

-85912 
-85963 
•86014 
•86064 
.86115 

1-16398 
1.16329 
1-16261 
1-16192 
1-16124 

•  66480 
•66501 
•66523 
•66545 
-66566 

.  74703 
.  74683 
•74664 
-  74644 
-74625 

•88992 

•89045 
•89097 
•89149 
•89201 

1-12369 
1.12303 
1.12238 
1.12172 
1-12106 

2€ 
18 
18 
17 
16 

45 
46 
47 
48 
49 

•65276 
-65298 
-65320 
-65342 
-65364 

.75756 
•75738 
.75719 
•75700 
.75680 

•86166 
86216 
.86267 
•86318 
-86368 

1-16056 
1-15987 
1-15919 
1.15851 
1-15783 

-66588 
-66610 
-66632 
-66653 
-66675 

-  74606 
-74586 
-74567 
•  74548 
•  74528 

•89253 
-89306 
-89358 
.89410 
•89463 

1.12041 
1-11975 
1.11909 
1.11844 
1-11778 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54 

-65386 
-65408 
.65430 
.65452 
•  65474 

.75661 
.75642 
•75623 
-75604 
•75585 

-86419 
•86470 
•86521 
•86572 
-.86623 

1-15715 
1-15647 
1-15579 
1-15511 
1-15443 

-66697 
-66718 
-66740 
-66762 
-66783 

•  74509 
-  74489 
•  74470 
•  74451 
-74431 

•89515 
•89567 
-89620 
-89672 
-89725 

1-11713 
1-11648 
1-11582 
1-11517 
1-11452 

10 

g 

8 

7 
6 

55 
56 
57 
58 
59 

•65496 
.65518 
.65540 
•65562 
.65584 

•75566 
•75547 
.75528 
•75509 
•75490 

.86674 
.86725 
.86776 
•86827 
•86878 

1-15375 
1-15308 
1.15240 
1-15172 
1-15104 

-66805 
-66827 
-66848 
.66870 
•66891 

•74412 
•74392 
•74373 
•74353 
•  74334 

•89777 
.89830 
-89883 
-89935 
-89988 

1.11387 
1-11321 
1-11256 
1-11191 
1-11126 

5 
4 
3 
2 

1 

6O 

.65606 

•75471 

-86929 

1-15037 

-66913 

•74314 

-  90040 

1-11061 

0 

1 

Cos. 

Sin. 

j  Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

49° 


48° 


NATURAL  SINES,   COSINES,   TANGENTS,   COTAN.     261 

42°  43° 


* 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

/ 

0 

2 
3 
4 

.66913 
66935 
.66956 
.66978 
.66999 

•74314 
•74295 
•74276 
•74256 
.  74237 

.90040 
•90093 
.90146 
.90199 
.90251 

1.11061 
1  10996 
1.10931 
1-10867 
1-10802 

68200 
-68221 
-68242 
-68264 
-68285 

•73135 
-73116 
-73096 
-73076 
-73056 

•93252 
•93306 
•93360 
•93415 
•93469 

1-07237 
1.07174 
1-07112 
1  •  07049 
1-06987 

6O 

59 
58 
57 
56 

5 
6 
7 
8 
9 

.67021 
67043. 
.67064 
.67086 
.67107 

.74217 
•74198 
74178 
•74159 
•74139 

!  90304 
.90357 
.90410 
•90463 
.90516 

1.10737 
1.10672 
1.10607 
1-10543 
1  .  10478 

68306 
-68327 
.68349 
.68370 
.68391 

•73036 
•73016 
•72996 
•72976 
-72957 

•93524 
•93578 
•93633 
-93688 
•93742 

1.06925 
1.06862 
1-06800 
1-06738 
1  06676 

55 
54 
53 
52 
51 

10 

11 
12 
13 
14 

.67129 
.67151 
.67172 
.67194 
•67215 

.74120 
.74100 
.  74080 
.74061 
.  74041 

.90569 
•90621 
.90674 
90727 
90781 

1-10414 
1.10349 
1-10285 
1-10220 
1.10156 

.68412 
-68434 
•68455 
•68476 
•68497 

-72937 
-72917 
-72897 
-72877 
-72857 

-93797 
-93852 
•93906 
-93961 
.94016 

1-06813 
1-06551 
1.06489 
1-06427 
1  06365 

5O 

49 
48 
47 
46 

15 
16 
17 
18 

19 

•67237 
.67258 
•67280 
67301 

67323 

.  74022 
•  74002 
•73983 
.73963 
•73944 

90834 
90887 
90940 
90993 
.91046 

1.10091 
1.10027 
1-09963 
1.09899 
1.09834 

•68518 
.68539 
•68561 
-68582 
.  68603 

.72837 
.72817 
-72797 
-72777 

•72757 

•94071 
-94125 
-94180 
•94235 
.94290 

1-06303 
1-06241 
1.06179 
1-06117 
1.06056 

45 
44 
43 
42 
41 

20 

21 
22 
23 
24 

67344 
67366 
67387 
67409 
67430 

.73924 
.  73904 
•  73885 

•73865 
•73846 

.91099 
91153 
91206 
91259 
91313 

1-09770 
1-09706 
1.09642 
1-09578 
1.09514. 

68624 
68645 
68666 
68688 
68709 

•  72737 
•72717 
-72697 
-72677 
.72657 

.94345 
-  94400 
-94455 
.94510 
.94565 

1-05994 
1-05932 
1.05870 
1-05809 
1-05747 

40 

39 
38 
37 
36 

25 
26 
27 
28 
29 

67452 
67473 
.67495 
•67516 
.67538 

.73826 
.73806 
.73787 
.73767 
•73747 

91366 
91419 
91473 
91526 
91580 

1-09450 
1.09386 
1-09322 
1-09258 
1.09195 

68730 
68751 
68772 
68793 
68814 

•72637 
•72617 
•72597 
•72577 
.72557 

.94620 
.94676 
-94731 
-94786 
-  94841 

1-05685 
1-05624 
1-05562 
1-05501 
1-05439 

35 
34 
33 
32 
31 

30 

31 
32 
33 

34 

.67559 
67580 
•67602 
•67623 
67645 

.73728 
-73708 
.73688 
.73669 
.74549 

91633 
91687 
91740 
91794 
91847 

1-09131 
1-09067 
1-09003 
1-08940 
1-08876 

68835 
68857 
68878 
68899 
68920 

•72537 
•72517 
-72497 
•72477 
.72457 

-94896 
-94952 
-95007 
-95062 
-95118 

1-05378 
1-05317 
1.05255 
1-05194 
1-05133 

30 

29 
28 
2? 
26 

35 
36 
37 
38 
39 

67666 
67688 
67709 
67730 
67752 

.73629 
.73810 
•73590 
73570 

.73551 

91901 
91955 
92008 
92062 

92116 

1-08813 
1-08749 
1-08686 
1-08622 
1  -08559 

68941 
68962 
68983 

69004 
69025 

.72437 
.72417 
.72397 
•72377 

-72357 

-95173 
•95229 
•95284 
-95340 
-95395 

1-05072 
1.05010 
1  -  04949 
1-04888 

1-04827 

25 
24 
23 
22 
21 

40 

41 
42 
43 
44 

-67773 
67795 
67816 
67837 
-67859 

.73531 
.73511 
.73491 
.73472 
.73452 

92170 
92224 
92277 
92331 
92385 

1-08498 
1-08432 
1-08369 
1-08306 
1  -  08243 

69046 
69067 
-69088 
-69109 
.69130 

•72337 
•72317 
•72297 
-72277 
.72257 

-95451 
•95506 
.95562 
.95618 
-95673 

1-04766 
1-04705 
1  •  04644 
1-04583 
1.04522 

20 

19 
18 
17 
16 

45 
16 
47 
48 
49 

-67880 
-67901 
-67923 
-  67944 
-67965 

-73432 
•73413 
•73393 
•73373 
•73353 

92439 
92493 
92547 
92601 
92655 

1.08179 
1-08116 
1-08053 
1-07990 
1-07927 

•69151 
•69172 
•69193 
•69214 
•69235 

-72236 
-72216 
•72196 
•72176 
•72156 

•95729 
.95785 
.95841 
.95897 
.95952 

1-04461 
1-04401 
1  -  04340 
1-04279 
1.04218 

15 
14 
13 
12 
11 

50 

51 
52 
53 
54_ 

•67987 
-68008 
-68029 
68051 
-68072 

•73333 
•73314 
•73294 
-73274 
•73254 

92709 
92763 
•92817 
.92872 
.92926 

1-07864 
1-07801 
1-07738 
1-07678 
1.07613 

-69258 
-69277 
•69298 
.69319 
-69340 

•72138 
•72116 
•72095 
.72075 
.72055 

.96008 
-96064 
-96120 
•96176 
-96232 

1-04158 
1-04097 
1.04036 
1-03978 
1-03915 

10 

9 
8 

_J 

55 
56 
57 
58 
59 

•68093 
•68115 
•68136 
•68157 
.68179 

.73234 
•73215 
•73195 
.73175 
.73155 

.92980 
•93034 
.93088 
•93143 
•93197 

1-07550 
1.07487 
1-07425 
1-07362 
1-07299 

-69361 
69382 
-69403 
69424 
-69445 

.72035 
.72015 
.71995 
•71974 
•71954 

-96288 
96344 
•96400 
96457 
.96513 

1.03855 
1-03794 
1-03734 
1-03874 
1-03613 

5 

1 

60 

.  68200 

.73135 

.93252 

1-07237 

69466 

71934 

.96569 

1  03553 

Q 

1 

Cos. 

Sin. 

Cot, 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

47° 


46° 


262    NATURAL  SINES,   COSINES,   TANGENTS,   COTAN. 

44°  44° 


' 

Sin. 

Cos. 

Tan. 

Cot, 

,  i 

/ 

Sin. 

Cos. 

Tan. 

Cot. 

/ 

0 

1 

2 

J_ 
5 
6 
7 
8 
9 

10 

11 
12 
13 

y_ 

15 
16 
17 
18 

12_ 
20 

21 
22 
23 
24_ 
25 
26 
27 
28 
29_ 

3<L 

.69466 
.69487 
.69508 
-69529 
.69549 
.  69570 
.69591 
•69612 
•69633 
.69654 
.69675 
.69696 
•69717 
•69737 
.69758 
•69779 
•69800 
.69821 
.69842 
.69862 

-71934 
.71914 
•71894 
•71873 
-71853 

•96569 
•96625 
•96681 
.96738 
.96794 

1-03553 
1.03493 
1.03433 
1.03372 
1.03312 

60 

59 
58 
57 
56 

3O 
31 

32 
33 
34 

.70091 
•70112 
•70132 
.70153 
.70174 
•70195 
.70215 
.70236 
•70257 
-70277 
-70298 
-70319 
-70339 
-70360 
-70381 

-71325 
-71305 
-71284 
•71264 
-71243 
•71223 
.71203 
-71182 
-71162 
.71141 
•71121 
-71100 
-  71080 
-71059 
-71039 

.98270 
-98327 
.98384 
•98441 
.98499 

1-01761 
1-01702 
1-01642 
1-01583 
1-01524 

3O 

29 
28 

27 
26 

•71833 
•71813 
.71792 
•71772 
.71752 

•96850 
•96907 
.96963 
•97020 
•97076 

1-03252 
1.03192 
1-03132 
1.03072 
1-03012 

55 
54 
53 
52 
51 

35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

•98556 
•98613 
-98671 
•98728 
.98786 
98843 
•98901 
•98958 
-99016 
-99073 

1-01465 
1-01406 
1-01347 
1-01288 
1.  01229 
1-01170 
1-01112 
1-01053 
1-00994 
1-00935 

25 
24 
23 
22 
21 
20 
19 
18 
17 
JL§ 
15 
14 
13 
12 
11 

•71732 
•71711 
.71691 
•71671 
.71650 

•97133 
•97189 
•97246 
•97302 
.97359 

1  02952 
1.02892 
1-02832 
-02772 
•02713 

50 

49 
48 
47 
46 

•71630 
•71610 
•71590 
•71569 

•71549 

•97416 
•97472 
.97529 
•97586 

•97643 

•02653 
.02593 
.02533 
•02474 
•02414 

45 
44 
43 
42 
41 

45 
46 
47 
48 
49 

-  70401 
-  70422 
70443 
-  70463 
-  70484 

-71019 
-70998 
.70978 
-70957 
70937 

-99131 
-99189 
•99247 
.99304 
.99362 

1-00876 
1-00818 
1-00759 
1-00701 
1-00642 

.69883 
-69904 
.69925 
.  69946 
.69966 
.69987 
•  70008 
•  70029 
.  70049 
.  70070 

•71529 
.71508 
•71488 
•71468 
•71447 

97700 
•97756 
•97813 
•97870 
•97927 

1-02355 
1.02295 
1.02236 
1-02176 
1-02117 
1.02057 
1.01998 
1-01939 
1.01879 
1-01820 

40 

39 
38 
37 
36 

50 

51 
52 
53 
54 

-70505 
.70525 
-70546 
-70567 
-  70587 

•70916 
•  70896 
•70875 
•  70855 
.  70834 

-99420 
-99478 
•99536 
-99594 
-99652 

1-00583 
1-00525 
1-00467 
1.00408 
1-00350 

10 
9 

8 

6 
~~| 

3 

2 
1 

.71427 
•71407 
.71386 
•71366 
•71345 
.  71325 

.97984 
98041 
.98098 
.98155 
•98213 

35 
34 
33 
32 
31 

55 
56 
57 
58 
59 

-  70608 
•70628 
•  70649 
-70670 
•  70890 

-70813 
-  70793 
-  70772 
-  70752 
-70731 

-99710 
-99768 
-99826 
-99884 
-99942 

1-00291 
1-00233 
1-00175 
1-00116 
1-00058 

•70091 

.98270 

1-01761 

30 

6O 

-70711 

.70711 

1  .  OOOOO 

1-00000 

0 

Cos. 

Sin. 

Cot. 

Tan. 

* 

' 

Cos. 

Sin. 

Cot. 

Tan. 

45° 


46° 


NATURAL  SECANTS  AND   COSECANTS 


263 


/ 

SECANTS. 

8 

0° 

1° 

2° 

3° 

4:° 

6° 

6° 

0 

1.00000 

1.00015 

1.00061 

1.00137 

1.00244 

1.00382 

-1.00551 

60 

1 

00000 

00016 

00062 

00139 

00246 

00385 

00554 

59 

2 

00000 

00016 

00063 

00140 

00248 

00387 

00557 

58 

3 

00000 

00017 

00064 

00142 

00250 

00390 

00560 

57 

4 

00000 

00017 

00065 

00143 

00252 

00392 

00563 

56 

5 

00000 

00018 

00066 

00145 

00254 

00395 

00566 

55 

6 

00000 

00018 

00067 

00147 

00257 

00397 

00569 

54 

7 

00000 

00019 

00068 

00148 

00259 

00400 

00573 

53 

8 

00000 

00020 

00069 

00150 

00261 

00403 

00576 

52 

9 

00000 

00020 

00070 

00151 

00263 

00405 

00579 

51 

10 

00000 

00021 

00072 

00153 

00265 

00408 

00582 

50 

11 

1.00001 

1.00021 

1.00073 

1.00155 

1.00267 

1.00411 

1.00585 

49 

12 

00001 

00022 

00074 

00156 

00269 

00413 

0058S 

48 

13 

00001 

00023 

00075 

00158 

00271 

00416 

00592 

47 

14 

00001 

00023 

00076 

00159 

00274 

00419 

00595 

46 

15 

00001 

00024 

00077 

00161 

00276 

00421 

00598 

45 

16 

00001 

00024 

00078 

00163 

00278 

00424 

00601 

44 

17 

00001 

00025 

00079 

00164 

00280 

00427 

00604 

43 

18 

00001 

00026 

00081 

00166 

00282 

00429 

00608 

42 

19 

00002 

00026 

00082 

00168 

00284 

00432 

00611 

41 

20 

00002 

00027 

00083 

00169 

00287 

00435 

00614 

40 

21 

1.00002 

1.00028 

1.00084 

1.00171 

1.00289 

1.00438 

1.00617 

39 

22 

00002 

00028 

00085 

00173 

00291 

00440 

00621 

38 

23 

00002 

00029 

00087 

00175 

00293 

00443 

00624 

37 

24 

00002 

00030 

00088 

00176 

00296 

00446 

00627 

36 

25 

00003 

00031 

00089 

00178 

00298 

00449 

00630 

35 

26 

00003 

00031 

00090 

00180 

00300 

00451 

00634 

34 

27 

00003 

00032 

00091 

00182 

00302 

00454 

00637 

33 

28 

00003 

00033 

00093 

00183 

00305 

00457 

00640 

3-2 

29 

00004 

00034 

00094 

00185 

00307 

00460 

00644 

31 

30 

00004 

00034 

00095 

00187 

00309 

00463 

00647 

30 

31 

1.00004 

1.00035 

1.00097 

1.00189 

1.00312 

1.00465 

1.00650 

29 

32 

00004 

00036 

00098 

00190 

00314 

00468 

00654 

28 

33 

00005 

00037 

00099 

00192 

00316 

00471 

00657 

27 

34 

00005 

00037 

00100 

00194 

00318 

00474 

00660 

26 

35 

00005 

00038 

00102 

00196 

00321 

00477 

00064 

25 

36 

00005 

00039 

00103 

00198 

00323 

00480 

00667 

24 

37 

00006 

00040 

00104 

00200 

00326 

00482 

00671 

23 

38 

00006 

00041 

00106 

00201 

00328 

00485 

00674 

22 

00006 

00041 

00107 

00203 

00330 

00488 

00677 

21 

40 

00007 

00042 

00108 

00205 

00333 

00491 

00681 

20 

41 

1.00007 

1.00043 

1.00110 

1.00207 

1.00335 

1.00494 

1.00684 

19 

42 

00007 

00044 

00111 

00209 

00337 

00497 

00688 

18 

43 

00008 

00045 

00113 

00211 

00340 

00500 

00691 

17- 

44 

00008 

00046 

00114 

00213 

00342 

00503 

00695 

16 

45 

00009 

00047 

00115 

00215 

00345 

00506 

00698 

15 

46 

00009 

00048 

00117 

00216 

00347 

00509 

00701 

14 

47 

00009 

00048 

00118 

00218 

00350 

00512 

00705 

13 

48 

00010 

00049 

00120 

00220 

00352 

00515 

00708 

12 

49 

00010 

00050 

00121 

00222 

00854 

00518 

00712 

11 

50 

00011 

00051 

00122 

00224 

00357 

00521 

00715 

10 

51 

1.00011 

1.00052 

1.00124 

1.00226 

1.00359 

1.00524 

1.00719 

9 

52 

00011 

00053 

00125 

00228 

00362 

00527 

00722 

8 

53 

00012 

00054 

00127 

00230 

00364 

00530 

00726 

7 

54 

00012 

00055 

•  00128 

00232 

00367 

00533 

00730 

6 

55 

00013 

00056 

00130 

00234 

00369 

00536 

00733 

5 

56 

00013 

00057 

00131 

00236 

00372 

00539 

00737 

4 

57 

00014 

00058 

00133 

00238 

00374 

00542 

00740 

3 

58 

00014 

00059 

00134 

00240 

00377 

00545 

00744 

2 

59 

00015 

00060 

00136 

00242 

00379 

.00548 

00747 

1 

60 

00015 

00061 

00137 

00244 

00382 

00551 

00751 

0 

/ 

89° 

88° 

87° 

86° 

86° 

84° 

83° 

/ 

COSECANTS. 

264 


NATURAL  SECANTS  AND  COSECANTS 


t 

SECANTS. 

/ 

i 

80 

9° 

10° 

11° 

12° 

13° 

0 

1.00751 

1.00983 

1.01247 

1.01543 

1.01872 

1.02234 

1.02630 

60 

i 

00755 

00987 

01251 

01548 

01877 

02240 

02637 

59 

2 

00758 

00991 

01256 

01553 

01883 

02247 

02644 

58 

3 

00762 

00995 

01261 

01558 

01889 

02253 

02651 

57 

4 

00765 

00999 

01265 

01564 

01895 

02259 

02658 

56 

5 

00769 

01004 

01270 

01569 

01901 

02266 

02665 

55 

6 

00773 

01008 

01275 

01574 

01906 

02272 

02672 

54 

7 

00776 

01012 

01279 

01579 

01912 

02279 

02679 

53 

8 

00780 

01016 

01284 

01585 

01918 

02285 

02686 

52 

9 

00784 

01020 

01289 

01590 

01924 

02291 

02693 

51 

10 

00787 

01024 

01294 

01595 

01930 

02298 

02700 

50 

11 

1.00791 

1.01029 

1.01298 

1.01601 

1.01936 

1.02304 

1.02707 

49 

12 

00795 

01033 

01303 

01606 

01941 

02311 

02714 

48 

13 

00799 

01037 

01308 

01611 

01947 

02317 

02721 

47 

14 

00802 

01041 

01313 

01616 

01953 

02323 

02728 

46 

15 

00806 

01046 

01318 

01622 

01959 

02330 

02735 

45 

16 

00810 

01050 

01322 

01627 

01965 

02336 

02742 

44 

17 

00813 

01054 

01327 

01633 

01971 

02343 

02749 

43 

18 

00817 

01059 

01332 

01638 

01977 

02349 

02756 

42 

19 

00821 

01063 

01337 

01643 

01983 

02356 

02763 

41 

•JO 

00825 

01067 

01342 

01649 

01989 

02362 

02770 

40 

21 

1.00828 

1.01071 

1.01346 

1.01654 

1.01995 

1.023G9 

1.02777 

39 

2-1 

00832 

01076 

01351 

01659 

02001 

02375 

02784 

38 

23 

00836 

01080 

01356 

01665 

02007 

02382 

02791 

37 

24 

00840 

01084 

01361 

01670 

02013 

02388 

02799 

30 

25 

00844 

01089 

01366 

01676 

02019 

02395 

02806 

35 

26 

00818 

01093 

01371 

01681 

02025 

02402 

02813 

34 

27 

00851 

01097 

01376 

01687 

02031 

02408 

02820 

33 

28 

00855 

01102 

01381 

01692 

02037 

02415 

02827 

32 

fe9 

00859 

01106 

01386 

01698 

0-2043 

02421 

02834 

31 

80 

00863 

01111 

01391 

01703 

02049 

02428 

02812 

30 

31 

1.00867 

1.01115 

1.01395 

1.01709 

1.02055 

1.02435 

1.02849 

29 

32 

00871 

01119 

01400 

01714 

02061 

02441 

02856 

28 

33 

00875 

01124 

01405 

01720 

02067 

02448 

028C3 

27 

34 

00878 

01128 

01410 

01725 

02073 

02454 

02870 

26 

35 

00882 

01133 

01415 

01731 

02079 

02461 

02878 

25 

36 

00886 

01137 

01420 

01736 

C2085 

02468 

02885 

24 

37 

00890 

01142 

01425 

01742 

020S1 

02474 

02892 

23 

38 

00894 

01146 

01430 

01747 

02097 

02481 

02899 

22 

39 

00898 

01151 

01435 

01753 

02103 

02488 

02907 

21 

40 

00902 

01155 

01440 

01758 

02110 

02494 

02914 

20 

41 

1.00906 

1.01160 

1.01445 

1.01764 

1.02116 

1.02501 

1.02921 

19 

42 

00910 

01164 

01450 

01769 

02122 

02508 

029-28 

18 

43 

00914 

01169 

01455 

01775 

02128 

02515 

02936 

17 

44 

00918 

01173 

01461 

01781 

02134 

02521 

02943 

16 

45 

00922 

01178 

01466 

01786 

02140 

02528 

02950 

15 

46 

00926 

01182 

01471 

01792 

02146 

02535 

02958 

14 

47 

00930 

01187 

01476 

01798 

02153 

02542 

02965 

13 

48 

00934 

01191 

01481 

01803 

02159 

02548 

02972 

12 

49 

00938 

01196 

01486 

01809 

02165 

02555 

02980 

11 

50 

00942 

01200 

01491 

01815 

02171 

02562 

02987 

10 

51 

1.00946 

1.01205 

1.01496 

1.01820 

1.02178 

1.02569 

1  .02994 

9 

52 

00950 

01209 

01501 

01826 

02184 

02576 

03002 

8 

53 

00954 

0121' 

01506 

01832 

02190 

02582 

03009 

7 

54 

00958 

01219 

01512 

01837 

02196 

02589 

03017 

6 

55 

00962 

01223 

01517 

01843 

02203 

02596 

03024 

5 

56 

00966 

01228 

01522 

01849 

02209 

02603 

03032 

4 

57 

00970 

01233 

015.27 

01854 

02215 

02610 

03039 

3 

58 

00975 

01237 

01532 

01860 

02221 

02617 

03046 

2 

59 

00979 

01242 

01537 

01866 

02228 

02624 

03054 

60 

00983 

01247 

01543 

01872 

02234 

02630 

03061 

0 

f 

82° 

81° 

80° 

79° 

78° 

7  jo 

76° 

i 

COSECANTS. 

NATURAL   SECANTS  AND   COSECANTS         265 


1 

SECANTS. 

/ 

14° 

15° 

16° 

17° 

18° 

19° 

20° 

0 

1.03061 

1.03528 

1.04030 

1.045G9 

1.05146 

1.05762 

1.06418 

60 

1 

03069 

03536 

04039 

04578 

05156 

05773 

06429 

59 

2 

03076 

03544 

04047 

04588 

051G6 

05783 

06440 

58 

3 

03084 

03552 

04056 

04597 

05176 

05794 

06452 

57 

4 

03091 

03560 

04065 

04606 

05186 

05805 

06463 

56 

5 

03099 

03568 

04073 

04616 

05196 

05815 

06474 

55 

6 

03106 

03576 

04082 

04625 

05206 

05826 

06486 

54 

7 

03114 

03584 

04091 

04635 

05216 

05836 

06497 

E3 

8 

03121 

03592 

04100 

04644 

05226 

05847 

06508 

52 

9 

03129 

03601 

0)108 

04653 

05236 

05858 

06520 

51 

10 

03137 

03609 

04117 

04663 

05246 

05869 

06531 

50 

11 

1.03144 

1.03617 

1.04126 

1.04672 

1.05256 

1  .05879 

1.06542 

49 

12 

03152 

03625 

04135 

04682 

0526(5 

05890 

06554 

48 

13 

03159 

03633 

04144 

04691 

05276 

05901 

06565 

47 

14 

03167 

03642 

04152 

04700 

05286 

05911 

06577 

46 

15 

03175 

03650 

04161 

04710 

05297 

05922 

06588 

45 

16 

03182 

03658 

04170 

04719 

05307 

05933 

06600 

44 

17 

03190 

03666 

04179 

04729 

05317 

05944 

06611 

4* 

18 

03198 

03674 

04188 

04738 

05327 

05955 

06622 

42 

19 

03205 

03683 

04197 

04748 

05337 

05965 

06634 

41 

20 

03213 

03691 

04206 

04757 

05347 

05976 

06645 

40 

21 

1.03221 

1.03699 

1.04214 

1.04767 

1.05357 

1.05987 

1.06657 

39 

22 

03228 

03708 

04223 

04776 

05367 

05998 

06668 

38 

23 

03236 

03716 

04232 

04786 

05378 

06009 

06680 

37 

24 

03244 

03724 

04241 

04795 

05388 

06020 

06691 

36 

25 

03251 

03732 

04250 

04805 

05398 

06030 

06703 

35 

26 

03259 

03741 

04259 

04815 

05408 

06041 

06715 

34 

27 

08267 

03749 

04268 

04824 

05418 

06052 

06726 

33 

28 

03275 

03758 

04277 

04834 

05429 

06063 

06738 

32 

29 

03282 

03766 

04286 

04843 

05439 

06074 

06749 

31 

30 

03290 

03774 

04295 

04853 

05449 

06085 

06761 

30 

31 

1.03298 

1.03783 

1.04304 

1.04863 

1.05460 

1.06096 

1.06773 

29 

32 

03306 

03791 

04313 

04872 

05470 

06107 

06784 

28 

w 

03313 

03799 

04322 

04882 

05480 

06118 

06796 

27 

34 

03321 

03808 

04331 

04891 

05490 

06129 

06807 

26 

35 

03329 

03816 

04340 

04901 

05501 

06140 

06819 

25 

36 

03337 

03825 

04349 

04911 

05511 

06151 

06831 

24 

37 

03345 

03833 

04358 

04920 

05521 

06162 

06843 

23 

33 

03353 

03842 

04367 

04930 

05532 

06173 

06854 

22 

39 

03360 

03850 

04376 

04940 

05542 

06184 

06866 

21 

40 

03368 

03858 

04385 

04950 

05552 

06195 

06878 

20 

41 

1.03376 

1.03867 

1.04394 

1.04959 

1.05563 

1.06206 

1.06889 

19 

42 

03384 

03875 

04403 

04969 

05573 

06217 

06901 

18 

43 

03392 

03884 

04413 

04979 

05584 

06228 

06913 

17 

44 

03400 

03892 

04422 

04989 

05594 

06239 

06925 

16 

45 

03408 

03901 

04431 

04998 

05604 

06250 

06936 

15 

46 

03416 

03909 

04440 

05008 

05615 

06261 

06948 

14 

47 

03424 

03918 

04449 

05018 

05625 

06272 

06960 

13 

48 

03432 

03927 

04458 

05028 

05636 

06283 

06972 

12 

.49 

03439 

03935 

04468 

05038 

05646 

06295 

06984 

11 

50 

03447 

03944 

04477 

05047 

05657 

06306 

06995 

10 

51 

1.03455 

1.03952 

1  .04486 

1  .05057 

1.05667 

1.06317 

1.07007 

9 

52 

03463 

03961 

04495 

05067 

05678 

06328 

07019 

8 

53 

03471 

03969 

04504 

05077 

05688 

06339 

07031 

7 

54 

03479 

03978 

01514 

05087 

05699 

06350 

07043 

6 

55 

03487 

03987 

04523 

05097 

05709 

06362 

07055 

5 

56 

03495 

03995 

04532 

05107 

05720 

06373 

07067 

4 

57 

03503 

04004 

04541 

05116 

05730 

06384 

07079 

3 

58 

03512 

04013 

04551 

05126 

05741 

06395 

07091 

2 

59 

03520 

04021 

04560 

05136 

05751 

06407 

07103 

1 

60 

03528 

04030 

04569 

05146 

05762 

06418 

07115 

0 

/ 

75° 

74° 

73° 

72° 

71° 

70° 

69° 

/ 

COSECANTS. 

266         NATURAL  SECANTS  AND  COSECANTS 


/ 

SECANTS. 

/ 

21° 

22° 

23° 

24° 

2o° 

26° 

27° 

0 

1.07115 

1.07853 

1.08636 

1.09464  1 

10338 

1.11260 

1.12233 

60 

1 

07126 

07866 

08649 

09478 

10353 

11276 

12249 

59 

2 

07138 

07879 

08663 

09492 

10368 

11292 

12266 

58 

3 

07150 

07892 

08676 

09506 

10383 

11308 

12283 

57 

4 

07162 

07904 

08690 

09520 

10398 

113-23 

12299 

56 

5 

07174 

07917 

08703 

09535 

10413 

11339 

12316 

55 

6 

07186 

07930 

08717 

09549 

10428 

11355 

12333 

54 

7 

07199 

07943 

08730 

09563 

10443 

11371 

12349 

53 

8 

07211 

07955 

08744 

09577 

10458 

11387 

12366 

52 

9 

07223 

07968 

08757 

09592 

10473 

11403 

12383 

51 

10 

07235 

07981 

08771 

09606 

10488 

11419 

12400 

50 

y 

1.07247 

1  .07994 

1.08784 

1.09620  1 

10503 

1.11435 

1.12416 

49 

12 

07259 

08006 

08798 

09635 

10518 

11451 

12433 

48 

13 

07271 

08019 

08811 

09649 

10533 

11467 

12450 

47 

14 

07283 

08032 

08825 

09663 

10549 

11483 

12467 

46 

15 

07295 

08045 

08839 

09678 

10564 

11499 

12484 

45 

16 

07307 

08058 

08852 

09692 

10579 

11515 

12501 

44 

17 

07320 

08071 

08866 

09707 

10594 

11531 

12518 

43 

18 

07332 

08084 

08880 

09721 

10609 

11547 

12534 

42 

19 

07344 

08097 

08893 

09735 

10625 

11563 

12551 

41 

20 

07356 

08109 

08907 

09750 

10640 

11579 

12568 

40 

21 

1.07368 

1.08122 

1.08921 

1.09764  1 

10655 

1.13595 

1.12585 

39 

22 

07380 

08135 

08934 

09779 

10670 

11611 

12602 

38 

23 

07393 

08148 

08948 

09793 

10686 

11627 

12619 

37 

24 

07405 

08161 

08962 

09808 

10701 

11643 

12636 

36 

25 

07417 

08174 

08975 

09822 

10716 

11659 

12653 

35 

26 

07429 

08187 

08989 

09837 

10731 

11675 

1267'0 

34 

27 

07442 

08200 

09003 

09851 

10747 

11691 

12687 

33 

28 

07454 

08213 

09017 

09866 

10762 

11708 

12704 

32 

29 

07466 

08226 

09030 

09880 

10777 

11724 

12721 

31 

30 

07479 

08239 

09044 

09895 

10793 

11740 

12738 

30 

31 

1.07491 

1.08252 

1.09058 

1.09909  1. 

10808 

1.11756 

1.12755 

29 

32 

07503 

08265 

09072 

09924 

10824 

11772 

12772 

28 

33 

07516 

08278 

09086 

09939 

10839 

11789 

12789 

27 

34 

07528 

08291 

09099 

09953 

10854 

11805 

12807 

26 

35 

07540 

08305 

09113 

09968 

10870 

11821 

188C4 

25 

36 

07553 

08318 

09127 

09982 

10885 

11838 

12841 

24 

37 

07565 

08331 

09141 

09997 

10901 

11854 

12858 

23 

38 

07578 

08344 

09155 

10012 

10916 

11870 

12875 

22 

39 

07590 

08357 

09169 

10026 

10932 

11886 

12892 

21 

40 

07602 

08370 

09183 

10041 

10947 

11903 

12910 

20 

41 

1.07615 

1.08383 

1.09197 

1.10055  1. 

10963 

1.11919 

1.12927 

19 

42 

07627 

08397 

09211 

10071 

10978 

11936 

12944 

18 

43 

07640 

08410 

09224 

10085 

10994 

11952 

12961 

17 

44 

07652 

08423 

09238 

10100 

11009 

11968 

12979 

16 

45 

07665 

08436 

09252 

10115 

11025 

11985 

12996 

15 

46 

07677 

08449 

09266 

10130 

11041 

12001 

13013 

14 

47 

07690 

08463 

09280 

10144 

11056 

12018 

13031 

13 

48 

07702 

08476 

09294 

10159 

11072 

12034 

13048 

12 

49 

07715 

08489 

09308 

10174 

11087 

12051 

13065 

11 

50 

07727 

08503 

09323 

10189 

11103 

12067 

13083 

10 

51 

1.07740 

1.08516 

1.09337 

1.10204  1. 

11119 

1.12084 

1.13100 

9 

52 

07752 

08529 

09351 

10218 

11134 

12100 

13117 

8 

53 

07765 

08542 

09365 

10233 

11150 

12117 

13135 

7 

54 

07778 

08556 

09379 

10248 

11166 

12133 

13152 

6 

55 

07790 

08569 

09393 

10263 

11181 

12150 

13170 

5 

56 

07803 

08582 

09407 

10278 

11197 

12166 

13187 

4 

57 

07816 

08596 

09421 

10293 

11213 

12183 

13205 

3 

58 

07828 

08609 

09435 

10308 

11229 

12199 

13222 

2 

59 

07841 

08623 

09449 

10323 

11244 

12216 

13240 

1 

60 

07853 

08636 

09464 

10338 

11260 

12233 

13257 

0 

/ 

68° 

67° 

66° 

65° 

64° 

63° 

62° 

/ 

COSECANTS. 

NATURAL  SECANTS  AND  COSECANTS 


267 


SECANTS.                      , 

28° 

29° 

30° 

31° 

32° 

33° 

34° 

0 

1  .  13257 

1.14335 

1.15470 

1.16663 

1.17918 

1.19236 

1.20622    60 

1 

13275 

14354 

15489 

16684 

17939 

19259 

2064: 

59 

2 

13292 

14372 

15509 

16704 

17961 

19281 

20669 

58 

3 

13310 

14391 

15528 

16725 

17982 

19304 

20693 

57 

4 

13327 

14409 

15548 

16745 

18004 

19327 

20717 

56 

5 

13345 

14428 

15567 

16766 

18025 

19349 

20740 

55 

6 

13362 

14446 

15587 

16786 

18047 

1937* 

20764 

54 

7 

13380 

14465 

15606 

16806 

18068 

19394 

20788 

53 

8 

13398 

14483 

15626 

16827 

18090 

19417 

20812 

52 

9 

13415 

14502 

15645 

16848 

18111 

19440 

20836 

51 

10 

13433 

14521 

15665 

16868 

18133 

19463 

20859 

50 

11 

1.13451 

1.14539 

1.15684 

1.16889 

1.18155 

1.19485 

1.20883 

49 

12 

13468 

14558 

15704 

16909 

18176 

19508 

20907 

48 

13 

13486 

14576 

15724 

16930 

18198 

19531 

20931 

47 

14 

13504 

14595 

15743 

16950 

18220 

19554 

20955 

46 

15 

13521 

14614 

15763 

16971 

18241 

19576 

20979 

45 

16 

13539 

14632 

15782 

16992 

18263 

19599 

21003 

44 

17 

13557 

14651 

15802 

17012 

18285 

19622 

21027 

43 

18 

13575 

14670 

15822 

17033 

18307 

19645 

21051 

42 

19 

13593 

14689 

15841 

17054 

18328 

19668 

21075 

41 

20 

13610 

14707 

15861 

17075 

18350 

19691 

21099 

40 

21 

1.13628 

1.14726 

1.15881 

1.17095 

1.18372 

1.19713 

1.21123 

39 

22 

13646 

14745 

15901 

17116 

18394 

19736 

21147 

38 

23 

13664 

14764 

15920 

17137 

18416 

19759 

21171 

37 

24 

13682 

14782 

15940 

17158 

18437 

19782 

21195 

36 

25 

13700 

14801 

15960 

17178 

18459 

19805 

21220 

35 

26 

13718 

14820 

15980 

17199 

18481 

19828 

21244 

34 

27 

13735 

14839 

16000 

17220 

18503 

19851 

21268 

33 

28 

13753 

14858 

16019 

17241 

18525 

19874 

21292 

32 

29 

13771 

14877 

16039 

17262 

18547 

19897 

21316 

81 

30 

13789 

14896 

16059 

17283 

18569 

19920 

21341 

30 

31 

1.13807 

1.14914 

1.16079 

1.17304 

1.18591 

\.19944 

1.21365 

29 

32 

13825 

14933 

16099 

17325 

18613 

19967 

21389 

28 

33 

13843 

14952 

16119 

17346 

18635 

19990 

21414 

27 

34 

13861 

14971 

16139 

17367 

18657 

20013 

21438 

26 

35 

13879 

14990 

16159 

17388 

18679 

20036 

21462 

25 

36 

13897 

15009 

16179 

17409 

18701 

20059 

21487 

24 

37 

18916 

15028 

16199 

17430 

18723 

20083 

21511 

23 

38 

13934 

15047 

16219 

17451 

18745 

20106 

21535 

22 

39 

13952 

15066 

16239 

17472 

18767 

20129 

21560 

21 

40 

13970 

15085 

16259 

17493 

18790 

20152 

21584 

20 

41 

1.13988 

1.15105 

1.16279 

1.17514 

1.18812 

1.20176 

1.21609 

19 

42 

14006 

15124 

16299 

17535 

18834 

20199 

21633 

18 

43 

14024 

15143 

16319 

17556 

18856 

20222 

21658 

17 

44 

14042 

15162 

16339 

17577 

18878 

20246 

21682 

16 

45 

14061 

15181 

16359 

17598 

18901 

20269 

21707 

15 

46 

14079 

15200 

16380 

17620 

18923 

20292 

21731 

14 

47 

14097 

15219 

16400 

17641 

18945 

20316 

21756 

13 

48 

14115 

15239 

16420 

17662 

18967 

20339 

21781 

12 

49 

14134 

15258 

16440 

17683 

18990 

20363 

21805 

11 

50 

14152 

15277 

16460 

17704 

19012 

20386 

21830 

10 

51 

1.14170 

1.15296 

1.16481 

1.17726 

1.19034 

1.20410 

1.21855 

9 

52 

14188 

15315 

16501 

17747 

19057 

20433 

21879 

8 

53 

14207 

15335 

16521 

17768 

19079 

20457 

21904 

7 

54 

14225 

15354 

16541 

17790 

19102 

20480 

21929 

6 

55 

14243 

15373 

16562 

17811 

19124 

20504 

21953 

5 

56 

14262 

15393 

16582 

17832 

19146 

20527 

21978 

4 

57 

14280 

15412 

16602 

17854 

19169 

20551 

22003 

8 

58 

14299 

15431 

16623 

17875 

19191 

20575 

22028 

2 

59 

14317 

15451 

16643 

17896 

19214 

20598 

22053 

1 

60 

14335 

15470 

16663 

17918 

19236 

20622 

22077 

0 

61° 

60° 

59° 

68° 

67° 

56° 

65° 

COSECANTS. 

NATURAL  SECANTS  AND   COSECANTS 


/ 

SECANTS. 

35° 

36° 

37° 

38° 

39° 

40° 

41° 

0 

1.22077 

1.23607 

1.25214 

1.26902 

1.28G76 

1.30541 

1.32501 

60 

1 

22102 

23633 

25241 

26931 

28706 

30573 

325:35 

59 

2 

22127 

23659 

25269 

26960 

28737 

30605 

32568 

58 

3 

22152 

23685 

25296 

26988 

28767 

30636 

32602 

57 

4 

22177 

23711 

25324 

27017 

28797 

30668 

32636 

56 

5 

22202 

23738 

25351 

27046 

28828 

30700 

32669 

55 

6 

22227 

23764 

25379 

27075 

28858 

30732 

32703 

54 

7 

22252 

23790 

25406 

27104 

28889 

30764 

32737 

53 

8 

22277 

23816 

25434 

27133 

28919 

30796 

32770 

52 

9 

22302 

23843 

25462 

27162 

28950 

30829 

32804 

51 

10 

22327 

23869 

25489 

27191 

28980 

30861 

32838 

50 

11 

1.22352 

1.23895 

1.25517 

1.27221 

1.29011 

1.30893 

1.32872 

49 

12 

22377 

23922 

25545 

27250 

29042 

30925 

32905 

48 

13 

22402 

23948 

25572 

27279 

29072 

30957 

82939 

47 

14 

22428 

23975 

25600 

27808 

29103 

30989 

32973 

46 

15 

22453 

24001 

25628 

27337 

29133 

31022 

33007 

45 

16 

22478 

24028 

25656 

27366 

29164 

31054 

33041 

44 

17 

22503 

24054 

25683 

27396 

29195 

31086 

33075 

43 

18 

22528 

24081 

25711 

27425 

29226 

31119 

33109 

42 

19 

22554 

24107 

25739 

27454 

29256 

31151 

33143 

41 

20 

22579 

24134 

25767 

27483 

29287 

31183 

33177 

40 

21 

1.22604 

1.24160 

1.25795 

1.27513 

1.29318 

1.31216 

1.33211 

39 

22 

22629 

24187 

25823 

27542 

29349 

31248 

33245 

38 

23 

22G55 

24213 

25851 

27572 

29380 

31281 

33279 

37 

24 

22680 

24240 

25879 

27601 

29411 

31313 

33314 

36 

25 

22706 

24267 

25907 

27630 

29442 

31346 

33348 

35 

26 

22731 

24293 

25935 

27660 

29473 

31378 

33382 

34 

27 

22756 

24320 

25963 

27689 

29504 

31411 

33416 

83 

28 

22782 

24347 

25991 

27719 

29535 

31443 

33451 

3? 

29 

22807 

24373 

26019 

27748 

29566 

31476 

33485 

31 

30 

22833 

24400 

26047 

27778 

29597 

31509 

3351  & 

30 

31 

1.22858 

1.24427 

1.26075 

1.27807 

1.29628 

1.31541 

1.33554 

29 

32 

22884 

24454 

26104 

27837 

29659 

31574 

33588 

28 

33 

22909 

24481 

26132 

27867 

29690 

31607 

33622 

27 

34 

22935 

24508 

26160 

27896 

29721 

31640 

33657 

26 

35 

22960 

24534 

26188 

27926 

29752 

31672 

33691 

25 

36 

22986 

24561 

26216 

27956 

29784 

31705 

33726 

24 

37 

23012 

24588 

26245 

27985 

29815 

31738 

33760 

23 

38 

23037 

24615 

26273 

28015 

29S46 

31771 

33795 

22 

39 

23063 

24642 

26301 

28045 

29877 

31804 

33830 

21 

40 

23089 

24669 

26330 

28075 

29909 

31837 

33864 

20 

41 

1.23114 

1.24696 

1.26358 

1.28105 

1.29940 

1.31870 

1.33899 

19 

42 

23140 

24723 

26387 

28134 

29971 

31903 

33934 

18 

43 

23166 

24750 

26415 

28164 

30003 

31936 

33968 

17  ! 

44 

23192 

24777 

26443 

28194 

30034 

31969 

34003 

16 

45 

23217 

24804 

26472 

28224 

30066 

32002 

34038 

15 

46 

23243 

24832 

26500 

28254 

30097 

32035 

3407S 

14 

47 

23269 

21859 

26529 

28284 

30129 

32068 

34108 

13 

48 

23295 

24886 

26557 

28314 

30160 

32101 

34142 

12 

49 

23321 

24913 

26586 

28344 

30192 

32134 

34177 

11 

50 

23347 

24940 

26615 

28374 

30223 

32168 

34212 

10 

51 

1.23373 

1.24967 

1.26643 

1.28404 

1.30255 

1.32201 

1.34247 

9 

52 

23399 

24995 

26672 

28434 

302.S7 

32234 

34282 

8 

53 

23424 

25022 

26701 

28464 

30318 

32267 

34317 

7  i 

54 

23450 

25049 

26729 

28495 

30350 

32301 

34352 

6 

55 

23476 

23077 

26758 

28525 

30382 

32334 

34387 

5 

56 

28502 

25104 

26787 

28555 

30413 

32368 

34423 

4 

57 

23529 

25181 

26815 

28585 

30445 

32401 

34458 

3 

58 

23555 

25159 

26844 

28615 

30477 

32434 

34498 

2 

59 

23581 

25186 

26873 

28646 

30509 

32468 

3452S 

1 

60 

23607 

25214 

26902 

28676 

30541 

32501 

34563 

0  ,' 

"t 

64° 

68° 

62° 

61° 

50° 

49° 

48° 

/   j 

COSECANTS. 

J 

NATURAL  SECANTS   AND   COSECANTS 


269 


f 

SECANTS. 

/ 

42° 

43° 

44° 

45° 

46° 

47° 

48° 

0 

1.34563 

1.36733 

1.39016 

1.41421 

1.43956 

1.46628 

1.49448 

60 

1 

34599 

36770 

39055 

41463 

43999 

46674 

49496 

59 

2 

34634 

36807 

39095 

41504 

44042 

46719 

49544 

58 

3 

34669 

36844 

39134 

41545 

44086 

46765 

49593 

57 

4 

34704 

36881 

39173 

41586 

44129 

46811 

49641 

56 

5 

34740 

36919 

39212 

41627 

44173 

46857 

49690 

55 

6 

34775 

36956 

39251 

41669 

44217 

46903 

49738 

54 

7 

34811 

36993 

39291 

41710 

44260 

46949 

49787 

53 

8 

34846 

37030 

39330 

41752 

44304 

46995 

49835 

52 

9 

34882 

37068 

39369 

41793 

44347 

47041 

49884 

51 

10 

34917 

37105 

39409 

41835 

44391 

47087 

49933 

50 

11 

1.34953 

1.37143 

1.39448 

1.41876 

1.44435 

1.47134 

1.49981 

49 

12 

34988 

37180 

39487 

41918 

44479 

47180 

50030 

48 

13 

35024 

37218 

39527 

41959 

44523 

47226 

50079 

47 

14 

35060 

37255 

39566 

42001 

44567 

47272 

50128 

46 

15 

35095 

37293 

39606 

42042 

44610 

47319 

50177 

45 

16 

35131 

37330 

39646 

42084 

44654 

47365 

50226 

44 

17 

35167 

37368 

39685 

42126 

44698 

47411 

50275 

43 

18 

35203 

37406 

39725 

42168 

44742 

47458 

50324 

42 

19 

35238 

37443 

39764 

42210 

44787 

47504 

50373 

41 

20 

35274 

37481 

39804 

42251 

44831 

47551 

50422 

40 

21 

1.35310 

1.37519 

1.39844 

1.42293 

1  44875 

1.47598 

1.50471 

39 

22 

35346 

37556 

39884 

42335 

44919 

47644 

50521 

38 

23 

35382 

37594 

39924 

42377 

44963 

47691 

50570 

37 

24 

35418 

37632 

39963 

42419 

45007 

47738 

50619 

36 

25 

35454 

37670 

40003 

42461 

45052 

47784 

50669 

35 

26 

35490 

37708 

40043 

42503 

45096 

47831 

50718 

34 

27 

35526 

37746 

40083 

42545 

45141 

47878 

50767 

33 

28 

35562 

37784 

40123 

42587 

45185 

47925 

50817 

32 

29 

35598 

37822 

40163 

42630 

45209 

47972 

50866 

31 

30 

35634 

37860 

40203 

42672 

45274 

48019 

50916 

30 

31 

1.35670 

1.37898 

1.40243 

1.42714 

1.45319 

1.48066 

1.50966 

29 

32 

35707 

37936 

40283 

42756 

45363 

48113 

51015 

28 

33 

35743 

37974 

40324 

42799 

45408 

48160 

51065 

27 

34 

35779 

38012 

40364 

42841 

45452 

48207 

51115    26 

35 

35815 

38051 

40404 

42883 

45497 

48254 

51165  !  25 

36 

35852 

38089 

40444 

42926 

45542 

48301 

51215    24 

37 

35888 

38127 

40485 

42968 

45587 

48349 

51265 

23 

38 

35924 

38165 

40525 

43011 

45631 

48396 

51314 

22 

39 

35961 

38204 

40565 

43053 

45676 

48443 

51364 

21 

40 

35997 

38242 

40606 

43096 

45721 

48491 

51415 

20 

41 

1.36034 

1.38280 

1.40646 

1.43139 

1.45766 

'.48538 

1.51465 

19 

42  |    36070 

38319 

40687 

43181 

45811 

48586 

51515 

18 

43     36107 

38357 

40727 

43224 

45856 

48633 

51565 

17 

44  i    36143 

38396 

40768 

43267 

45901 

48681 

51615 

16 

45 

36180 

38434 

40808 

43310 

45946 

48728 

51665 

15 

46 

36217 

38473 

40849 

43352 

45992 

48776 

51716 

14 

47 

36253 

38512 

40890 

43395 

46037 

48824 

51766 

13 

48 

36290 

38550 

40930 

43438 

46082 

48871 

51817 

12 

49 

36327 

38589 

40971 

43481 

46127 

48919 

51867 

11 

50 

36363 

38628 

41012 

43524 

46178 

48967 

51918 

10 

51 

1.36400 

1.38666 

1.41053 

1.43567 

1.46218 

1.49015 

1.51968 

9 

52 

36437 

38705 

41093 

43610 

46263 

49063 

52019 

8 

53 

36474 

38744 

41134 

43653 

46309 

4911'! 

52069 

7 

54 

36511 

38783 

41175 

43696 

46354 

49159 

52120 

6 

55 

36548 

38822 

41'.'16 

43739 

46400 

49207 

52171 

5 

56 

36585 

38860 

41257 

43783 

46445 

49255 

52222 

4 

57 

36622 

38899 

41298 

43826 

46491 

49303 

52273 

3 

58 

36659 

38938 

41339 

43869 

46537 

49351 

52323 

2 

59 

36696 

38977 

41380 

43912 

46582 

49399 

52374 

1 

60 

36733 

39016 

41421 

43956 

46628 

49448 

52425 

0 

/ 

47° 

46° 

45° 

44° 

43° 

42° 

41° 

/ 

COSECANTS. 

270 


NATURAL  SECANTS   AND   COSECANTS 


/ 

SECANTS. 

/ 

49° 

50° 

61° 

62° 

53° 

54° 

55° 

0 

1.52425 

1.55572 

1.58902 

1.62427 

1.66164 

1.70130 

1.74345 

60 

1 

52476 

55626 

58959 

62487 

66228 

70198 

74417 

59 

2 

52527 

55680 

59016 

62548 

66292 

70267 

74490 

58 

3 

52579 

55734 

59073 

62609 

66357 

70335 

74562 

57 

4 

52630 

55789 

59130 

62669 

66421 

70403 

74635 

56 

6 

52681 

55843 

59188 

62730 

66486 

70472 

74708 

55 

6 

52732 

55897 

59245 

62791 

66550 

70540 

74781 

54 

7 

52784 

55951 

59302 

62852 

66615 

70609 

74854 

53 

8 

52835 

56005 

59360 

62913 

66679 

70677 

74927 

52 

9 

52886 

56060 

59418 

62974 

66744 

70746 

75000 

51 

10 

52938 

56114 

59475 

63035 

66809 

70815 

75073 

50 

11 

1.52989 

1.56169 

1.59533 

1.63096 

1.66873 

1.70884 

1.75146 

49 

12 

53041 

56223 

59590 

63157 

66938 

70953 

75219 

48 

13 

53092 

56278 

59648 

63218 

67003 

71022 

75293 

47 

14 

53144 

56332 

59706 

63279 

67068 

71091 

75366 

46 

15 

53196 

56387 

59764 

63341 

67133 

71160 

75440 

45 

16 

53247 

56442 

59822 

63402 

67199 

71229 

75513 

44 

17 

53299 

56497 

59880 

63464 

67264 

71298 

75587 

43 

18 

53351 

56551 

59938 

63525 

67329 

71368 

75661 

42 

19 

53403 

56606 

59996 

63587 

67394 

71437 

75734 

41 

20 

53455 

56661 

60054 

63648 

67460 

71506 

75808 

40 

21 

1.53507 

1.56716 

1.60112 

1.63710 

1.67525 

1.71576 

1.75882 

39 

22 

53559 

56771 

60171 

63772 

67591 

71646 

75956 

38 

23 

53611 

56826 

60229 

63834 

07656 

71715 

76031 

37 

24 

53663 

56881 

602S7 

63895 

67722 

71785 

76105 

36 

25 

53715 

56937 

60346 

63957 

67788 

71855 

76179 

35 

26 

53768 

56992 

60404 

64019 

67853 

71925 

76253 

34 

27 

53820 

57047 

604G3 

64081 

67919 

71995 

76328 

S3 

28 

53872 

57103 

60521 

64144 

67985 

72065 

76402 

32 

29 

53924 

57158 

60580 

64206 

68051 

72135 

76477 

31 

30 

53977 

57213 

60639 

64268 

68117 

72205 

76552 

30 

31 

1.540-29 

1.57269 

1.60698 

1.64330 

1.68183 

1.72275 

1.76626 

29 

32 

54082 

5732  1 

60756 

64393 

68250 

72346 

76701 

28 

33 

54134 

57380 

60815 

64455 

68316 

72416 

76776 

27 

34 

54187 

57436 

60874 

64518 

68382 

72487 

76851 

26 

35 

51240 

57491 

60933 

64580 

68449 

72557 

76926 

25 

36 

54292 

57547 

60992 

64643 

68515 

72628 

77001 

24 

37 

54345 

57603 

61051 

64705 

68582 

72698 

77077 

23 

38 

54398 

57659 

61111 

64768 

68648 

72769 

77152 

22 

39 

54451 

57715 

61170 

64831 

68715 

72840 

77227 

21 

40 

54504 

57771 

61229 

64894 

6878? 

72911 

77303 

20 

41 

1.54557 

1.57827 

1.61288 

1.64957 

1.68848 

1.72982 

1.77378 

19 

42 

54610 

57883 

61348 

65020 

68915 

73053 

77454 

18 

43 

54663 

57939 

61407 

65083 

68982 

73124 

77530 

17 

•  44 

54716 

57995 

61467 

65146 

69049 

73195 

77606 

16 

45 

54769 

58051 

61526 

65209 

69116 

73267 

77681 

15 

46 

54822 

58108 

61586 

65272 

69183 

73338 

77757 

14 

47 

54876 

58164 

61646 

65336 

69250 

73409 

77833 

13 

48 

54929 

58221 

61705 

65399 

69318 

73481 

77910 

12 

49 

54982 

58277 

61765 

65462 

69385 

73552 

77986 

11 

50 

55036 

58333 

61825 

65526 

69452 

73624 

78062 

10 

51 

1.55089 

1.58390 

1.61885 

1.65589 

1.69520 

1.73696 

1.78138 

9 

52 

55143 

58447 

61945 

65653 

69587 

73768 

78215 

8 

53 

55196 

58503 

62005 

65717 

69655 

73840 

78291 

7 

54 

55250 

58560 

62065 

65780 

69723 

73911 

78368 

6 

55 

55803 

58617 

62125 

65844 

69790 

73983 

78445 

5 

56 

55357 

58674 

62185 

65908 

69858 

74056 

78521 

4 

57 

55411 

58731 

62246 

65972 

69926 

74128 

78598 

3 

58 

55465 

58788 

62306 

66036 

69994 

74200 

78675 

2 

59 

55518 

58845 

62366 

66100 

70062 

74272 

78753 

1 

60 

55572 

58902 

62427 

66164 

70130 

74345 

78829 

0 

| 

40« 

89° 

38° 

87° 

86° 

86° 

84° 

/ 

COSECANTS. 

( 

NATURAL  SECANTS  AND   COSECANTS         271 


/ 

SECANTS. 

/ 

56° 

57° 

58° 

59° 

60° 

61° 

62° 

0 

1.78829 

1.83608 

1.88708 

1.94160 

2.00000 

2.06267 

2.13005 

60 

1 

789C6 

83690 

88796 

94254 

00101 

06375 

13122 

59 

o 

78984 

83773 

88884 

94349 

00202 

06483 

13239 

58 

3 

79061 

83855 

88972 

94443 

00303 

06592 

13356 

57 

4 

79138 

83938 

89060 

94537 

00404 

06701 

13473 

56 

5 

79216 

84020 

89148 

94632 

00505 

06809 

13590 

55 

6 

79293 

84103 

89237 

94726 

00607 

06918 

13707 

54 

7 

79371 

84186 

89325 

94821 

00708 

07027 

13825 

53 

8 

79449 

84269 

89414 

94916 

00810 

07137 

13942 

52 

9 

79527 

84352 

89503 

95011 

00912 

07246 

14060 

51 

10 

79604 

84435 

89591 

95106 

01014 

07356 

14178 

50 

n 

1.79682 

1.84518 

1.89680 

1.95201 

2.01116 

2.07465 

2.14296 

49 

12 

79761 

84601 

89769 

95296 

01218 

07575 

14414 

48 

13     79839 

84685 

89858 

95392 

01320 

07685 

14533 

47 

14     79917 

84768 

89948 

95487 

01422 

07795 

14651 

46 

15     79995 

84852 

90037 

95583 

01525 

07905 

14770 

45 

16     80074 

84935 

90126 

95678 

01628 

08015 

14889 

44 

17     80152 

85019 

90216 

95774 

01730 

08126 

15008 

43 

18 

80231 

85103 

90305 

95870 

01833 

08236 

15127 

42 

19 

80309 

85187 

90395 

95966 

01936 

08347 

15246 

41 

20     80388 

85271 

90485 

96062 

02039 

08458 

15366 

40 

21 

L  80467 

1.85355 

1.90575 

1.96158 

2.02143 

2  08569 

2.15485 

39 

22 

80546 

85439 

90665 

96255 

02246 

08680 

15605 

38 

23  i    80625 

85523 

90755 

96351 

02349 

08791 

15725 

37 

24     80704 

85608 

90845 

96448 

02453 

08903 

15845 

36 

25 

80783 

85692 

90935 

96544 

02557 

09014 

15965 

35 

26 

80862 

85777 

91026 

96641 

02661 

09126 

16085 

34 

27 

80942 

85861 

91116 

96738 

02765 

09238 

16206 

33 

28 

81021 

85946 

91207 

96835 

02869 

09350 

16326 

32 

29 

81101 

86031 

91297 

96932 

02973 

09462 

16447 

31 

30 

81180 

86116 

91388 

97029 

03077 

09574 

16568 

30 

31 

1.81260 

1  .86201 

1.91479 

1.97127 

2.03182 

2.09686 

2.16689 

29 

82 

81340 

86286 

91570 

97224 

03286 

09799 

16810 

28 

33 

81419 

86371 

91661 

97322 

03391 

09911 

16932 

27 

34 

81499 

86457 

91752 

97420 

03496 

10024 

17053 

26 

35 

81579 

86542 

91844 

97517 

03601 

10137 

17175 

25 

36 

81659 

86627 

91935 

97615 

03706 

10250 

17297 

24 

37 

81740 

86713 

92027 

97713 

03811 

10363 

17419 

23 

38 

81820 

86799 

92118 

97811 

03916 

10477 

17541 

22 

39 

81900 

86885 

92210 

97910 

04022 

10590 

17663 

21 

40 

81981 

86990 

92302 

98008 

04128 

10704 

17786 

20 

41 

1.82061 

1  .87056 

1.92394 

1.98107 

2.04233 

2.10817 

2.17909 

19 

42 

82142 

87142 

92486 

98205 

04339 

10931 

18031 

18 

43 

82222 

87229 

92578 

98304 

04445 

11045 

18154 

17 

44 

82303 

87315 

92670 

98403 

04551 

11159 

18277 

16 

45 

82384 

87401 

92762 

98502 

04658 

11274 

18401 

15 

46 

82465 

87488 

92855 

98601 

04764 

11388 

18524 

14 

47 

82546 

87574 

92947 

98700 

04870 

11503 

18648 

13 

48 

82627 

87661 

93040 

98799 

04977 

11617 

18772 

12 

49 

82709 

87748 

93133 

98899 

05084 

11732 

18895 

11 

50 

82790 

87834 

93226 

98998 

05191 

11847 

19019 

10 

51 

1.82871 

1.87921 

1.93319 

1.99098 

2.05298 

2.11963 

2.19144 

9 

52 

82953 

88008 

93412 

99198 

05405 

12078 

19268 

8 

53 

83034 

88095 

93505 

99298 

05512 

12133 

19393 

7 

54 

83116 

88183 

93598 

99398 

05619 

12309 

19517 

6 

55 

83198 

88270 

93692 

99498 

05727 

12425 

19642 

5 

56 

83280 

88357 

93785 

99598 

05835 

12540 

19767 

4 

57 

83362 

88445 

93879 

99698 

05942 

12657 

19892 

3 

58 

83444 

88532 

93973 

99799 

06050 

12773 

20018 

2 

59 

83526 

88620 

94066 

99899 

06158 

12889 

20143 

1 

60 

83608 

88708 

94160 

2.00000 

06267 

13005 

20269 

0 

/ 

33° 

32° 

31° 

30° 

29° 

28° 

27° 

/ 

COSECANTS. 

272 


NATURAL  SECANTS  AND  COSECANTS 


! 
, 

SECANTS. 

/ 

63° 

64° 

65° 

66° 

67° 

68° 

69° 

0 

2.20269 

2.28117 

2.36620 

2.45859  2.55930 

2.66947 

2.79043 

60 

1 

20395 

28-253 

367C8 

460-20 

56106 

67139 

79254 

59 

2 

20521 

28390 

36910 

46181 

56-282 

67332 

79460 

58 

3 

20647 

28526 

37064 

46342 

56458 

675-25 

79679 

57 

4 

20773 

28663 

37212 

46504 

56634 

67718 

79891 

56 

5 

20900 

28800 

37361 

46665 

56311 

67911 

80104 

55 

6 

21026 

28937 

37509 

46827 

56988 

68105 

80318 

54 

7 

21153 

29074 

37658 

46989 

57165 

68299 

80531 

53 

8 

21280 

29211 

37808 

47152 

57342 

68494 

80746 

52 

9 

21407 

29349 

37957 

47314 

575-JO 

68689 

80960 

51 

10 

21535 

29487 

38107 

47477 

57698 

68884 

81175 

50 

11 

2.21662 

2.29625 

2.38256 

2.47640  2 

57876 

2.69079 

a.  81390 

49 

12 

21790 

29763 

38406 

47804 

58054 

69275 

81605 

48 

13 

21938 

29901 

38556 

479G7 

58333 

69471 

81821 

47 

14 

22045 

30040 

38707 

48131 

58412 

69667 

82037 

46 

15 

22174 

30179 

38857 

48295 

58591 

69864 

8-2254 

45 

16 

22302 

30318 

39008 

48459 

58771 

70061 

82471 

44 

17 

22430 

30457 

39159 

48624 

58950 

70258 

82688 

43 

18 

22559 

30596 

39311 

48789 

59130 

70455 

82906 

42 

19 

22688 

30735 

39462 

48954 

59311 

70653 

831-24 

41 

20 

22817 

30875 

39614 

49119 

59491 

70851 

83342 

40 

21 

2.22946 

2  31015 

2.39766 

2.49284  2 

59672 

2.71050 

2.83561 

39 

22 

23075 

31155 

39918 

49450 

59853 

71249 

83780 

38 

23 

23205 

31295 

40070 

49616 

60035 

71448 

83999 

37 

24 

23334 

31436 

40222 

49782 

60-217 

71647 

84219 

36 

25 

23464 

31576 

40375 

49948 

60399 

71847 

84439 

35 

26 

2359  1 

31717 

40528 

50115 

60581 

72047 

84059 

34 

27 

39724 

31858 

40681 

50282 

60763 

72247 

84S80 

33 

28 

23855 

31999 

40835 

50449 

60946 

72448 

85102 

32 

29 

23985 

32140 

409F8 

50617 

61129 

72649 

853-23 

31 

30 

24116 

32282 

41112 

50784 

61313 

72850 

85545 

30 

31 

2.24247 

2.32424 

2  41296 

2.50952  2 

61496 

2.73052 

2.85767 

29 

32 

24378 

32566 

41450 

51120 

61680 

73254 

85990 

28 

Jffl 

24509 

32708 

41605 

51289 

61864 

73456 

86213 

27 

34 

24640 

32850 

41760 

51457 

62049 

73659 

86437 

26 

35 

24772 

32993 

41914 

51626 

62234 

73862 

86661 

25 

36 

24903 

33135 

42070 

51795 

62419 

74065 

86885 

24 

37 

25035 

33278 

42225 

51965 

62604 

74269 

87109 

23 

38 

25167 

33422 

42380 

52134 

62790 

74473 

87334 

22 

39 

25300 

33565 

42536 

52304 

62976 

74677 

87560 

21 

40 

25432 

33708 

42692 

52474 

63162 

74881 

87785 

20 

41 

2.25565 

2.33852 

2.42848 

2.52645  2 

63348 

2.75086 

2.88011 

19 

42 

25697 

33996 

43005 

52815 

63535 

75292 

88238 

18 

43 

25830 

34140 

43162 

52986 

63722 

75497 

88465 

17 

44 

25963 

31284 

43318 

53157 

63909 

75703 

88692 

16 

45 

26097 

34429 

43476 

53329 

64097 

75909 

889-20 

15 

46 

26230 

34573 

43633 

53500 

64-285 

76116 

89148 

14 

47 

26364 

34718 

43790 

5367-2 

64473 

76323 

89376 

13 

48 

26498 

34863 

43948 

53845 

64662 

76530 

89605 

12 

49 

26632 

35009 

44106 

54017 

64851 

76737 

89834 

11 

50 

26766 

35154 

44264 

54190 

65040 

76945 

90003 

10 

51 

2.26900 

2.35300 

2.44423 

2  54363  2.65229 

2.77154 

2.90-293 

9 

52 

27035 

35446 

44582 

54536 

65419 

77362 

90524 

8 

53 

27169 

35593 

41741 

54709 

65609 

77571 

90754 

7 

54 

27304 

35738 

44900 

54883 

65799 

77780 

90986 

6 

55 

27439 

35885 

45059 

55057 

65989 

77990 

91217 

5 

56 

27574 

36031 

45219 

55231 

66180 

78200 

91449 

4 

57 

27710 

36178 

45378 

55405 

66371 

78410 

91681 

3 

58 

27845 

36325 

45539 

55580 

66563 

78621 

91914 

2 

59 

27981 

36473 

45699 

55755 

66755 

7883-2 

92147 

1 

60 

28117 

36620 

45859 

55930 

66947 

79043 

92380 

0 

26° 

25° 

24° 

28° 

22° 

21° 

20° 

/ 

COSECANTS. 

NATURAL  SECANTS   AND   COSECANTS 


273 


1 

SECANTS. 

/ 

70° 

71° 

72° 

73° 

74° 

75° 

?6° 

0 

2.92380 

3.07155 

3.23607 

3.42030  3 

.62796 

3.86370 

4.13357 

60 

1 

92614 

07415 

23897 

42356 

63164 

86V90 

13839 

59 

2 

92849 

07675 

24187 

42683 

63533 

87211 

14323 

58 

3 

93083 

07936 

24478 

43010 

63903 

876:33 

14809 

57 

4 

93318 

08197 

24770 

43337 

64274 

88056 

15295 

56 

5 

93554 

08459 

25062 

43666 

64645 

88479 

15782 

55 

6 

93790 

08721 

25355 

43995 

65018 

88904 

16271 

54 

7 

94026 

08983 

25648 

44324 

65391 

89330 

16761 

53 

8 

94263 

09246 

25942 

44655 

65765 

89756 

17252 

52 

9 

94500 

09510 

26237 

44986 

66  MO 

90184 

17744 

51 

10 

94737 

09774 

26531 

45317 

66515 

900  13 

18238 

50 

11 

2.94975 

3.10038 

3.26827 

3.45650  3 

.66892 

3.9'U42 

4.18733 

49 

12 

95213 

10303 

27123 

45983 

67269 

9  1473 

19228 

43 

13 

95452 

10568 

27420 

46316 

67647 

91904 

19725 

47 

14 

95(591 

10834 

27717 

46651 

68025 

92337 

20224 

46 

15 

95931 

11101 

28015 

46986 

68405 

92770 

20723 

45 

16 

96171 

11  867 

28313 

47321 

68785 

93204 

21224 

44 

17 

96411 

11035 

28612 

47658 

69167 

93640 

21726 

43 

18 

96652 

11903 

28912 

47995 

69549 

94076 

22229 

42 

19 

96^93 

12171 

29212 

48333 

69931 

94514 

22734 

41 

20 

97135 

12440 

29512 

48671 

70315 

94952 

23239 

40 

21 

2.97377 

3.12709 

3.29814 

3.49010  3 

.70700 

3.95392 

4.23746 

39 

22 

97619 

12979 

30115 

49350 

71085 

95832 

24255 

38 

23 

97862 

13249 

30418 

49691 

71471 

96274 

24764 

37 

24 

98106 

13520 

30721 

50032 

71858 

96716 

25275 

36 

25 

98349 

13791 

31024 

50374 

72246 

97160 

25787 

35 

26 

98594 

14063 

31328 

50716 

72635 

97604 

26300 

34 

27 

98838 

14335 

31633 

51060 

73024 

98050 

26814 

33 

28 

99083 

14608 

31939 

51404 

73414 

98497 

27330 

32 

29 

99329 

14881 

32244 

51748 

73806 

98944 

27847 

31 

30 

99574 

15155 

32551 

52094 

74198 

99393 

28366 

30 

31 

2.99821 

3.15429 

3.32858 

3.52440  3 

.74591 

3.99843 

4.28885 

29 

82 

3.00067 

15704 

33166 

52787 

74984 

4.00293 

29406 

28 

33 

00315 

15979 

33474 

53134 

75379 

00745 

29929 

27 

34 

00562 

16255 

33783 

53482 

75775 

01198 

30452 

26 

35 

00810 

16531 

34092 

53831 

76171 

01652 

30977 

25 

36 

01059 

16808 

34403 

54181 

76568 

02107 

31503 

24 

3V 

01308 

17085 

34713 

54531 

76966 

025G3 

32031 

23 

38 

01557 

17363 

35025 

54883 

77365 

03020 

32560 

22 

39 

01807 

17641 

35336 

55235 

77765 

03479 

33090 

21 

40 

02057 

17920 

35649 

55587 

78166 

03938 

33622 

20 

41 

3.02308 

3.18199 

3.35962 

3.55940  3 

.78568 

4.04398 

4.34154 

19 

42 

02559 

18479 

36276 

56294 

78970 

04860 

34689 

18 

43 

02810 

18759 

36590 

56649 

79374 

05322 

35224 

17 

44 

03062 

19040 

36905 

57005 

79778 

05786 

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16 

45 

03315 

19322 

37221 

57361 

80183 

06251 

36299 

15 

46 

03568 

19604 

37537 

57718 

80589 

06717 

36839 

14 

47 

03821 

19886 

37854 

58076 

80996 

07184 

37380 

13 

48 

04075 

20169 

38171 

58434 

81404 

07652 

37923 

12 

49 

04329 

20453 

38489 

58794 

81813 

08121 

38466 

11 

50 

04584 

20737 

38808 

59154 

82223 

08591 

39012 

10 

51 

3.04839 

3.21021 

3.39128 

3.59514  3 

.82633 

4.09063 

4.39558 

9 

52 

05094 

21306 

39448 

59876 

83045 

09535 

40106 

8 

53 

05350 

21592 

39768 

60238 

83457 

10000 

40656 

7 

54 

05607 

21878 

40089 

60601 

83871 

10484 

41206 

6 

55 

05864 

22165 

'  40411 

60965 

84285 

10960 

41759 

5 

56 

06121 

22452 

40734 

61330 

84700 

11437 

42312 

4 

57 

06379 

22740 

41057 

61695 

85116 

11915 

42867 

3 

58 

06637 

23028 

41381 

62061 

85533 

12394 

43424 

2 

59 

06896 

23317 

41705 

62428 

85951 

12875 

43982 

1 

60 

07155 

23607 

42030 

62796 

86370 

13357 

44541 

0 

~~^~~ 

/ 

19° 

18° 

17° 

16° 

16° 

14° 

18° 

t 

COSECANTS. 

274         NATURAL  SECANTS  AND  COSECANTS 


/ 

SECANTS. 

/ 

77' 

78° 

79° 

80° 

81° 

82° 

88° 

0 

4.44541 

4.80973 

5.24084 

5.75877  6 

.39245 

7.18530 

8.20551 

60 

1 

45102 

81633 

24870 

76829 

40422 

5:0020 

22500 

59 

2 

45664 

82294 

25658 

77784 

41602 

21517 

24457 

58 

8 

46228 

82956 

26448 

78742 

42787 

23019 

26425 

57 

4 

46793 

83621 

27241 

79703 

43977 

24529 

28402 

56 

5 

47360 

84288 

28036 

80667 

45171 

26044 

30388 

55 

6 

47928 

84956 

28833 

81635 

46369 

27566 

32384 

54 

7 

48498 

85627 

29634 

83606 

47572 

29095 

34390 

53 

8 

49069 

86299 

30436 

83581 

48779 

30630 

36405 

52 

9 

49642 

86973 

31241 

84558 

49991 

32171 

38431 

51 

10 

50216 

87649 

32049 

85539 

51208 

33719 

40466 

50 

11 

4.50791 

4.88327 

5.32859 

5.86524  6 

.52429 

7.35274 

8.42511 

49 

12 

51368 

89007 

33G7  1 

87511 

53655 

36835 

44566 

48 

13 

51947 

89689 

34486 

88502 

54886 

3S403 

46632 

47 

14 

52527 

90373 

35304 

89497 

56121 

39978 

48707 

46 

15 

53109 

91058 

36124 

90495 

57361 

41560 

50793 

45 

16 

53692 

91746 

36947 

91496 

58606 

43148 

52889 

44 

17 

54277 

92436 

37772 

92501 

59855 

44743 

54996 

43 

18 

54663 

93128 

38600 

93509 

61110 

46346 

571L3 

42 

19 

55451 

93821 

39430 

94521 

62369 

47955 

59241 

41 

20 

56041 

94517 

402U3 

95536 

63033 

49571 

61379 

40 

21 

4.56632 

4.95215 

5.41099 

5.9C555  6 

.64902 

7.51194 

8.63528 

39 

22 

57224 

95914 

41937 

97577 

66176 

52825 

65688 

38 

23 

57819 

96616 

42778 

98603 

67454 

54462 

67859 

37 

24 

58414 

97320 

43622 

90633 

68738 

56107 

70041 

36 

25 

59012 

98025 

44468 

6.00666 

70027 

57759 

72234 

35 

26 

59611 

98733 

45317 

01703 

71321 

59418 

74438 

34 

27 

60211 

99443 

46169 

02743 

7-J620 

61085 

76653 

33 

28 

60813 

5.00155 

47023 

03787 

73924 

62759 

78880 

32 

29 

61417 

00869 

47881 

04834 

75233 

64441 

81118 

31 

30 

62023 

01585 

48T40 

05886 

76547 

66130 

83367 

30 

31 

4  62630 

5.02303 

5.49603 

6.06941  6 

.77866 

7.67826 

8.85628 

29 

32 

63238 

03024 

5046S 

08000 

79191 

69530 

87901 

28 

33 

63849 

03746 

51337 

09062 

80521 

71242 

90186 

27 

34 

64461 

04471 

52208 

10129 

81856 

72962 

92482 

26 

35 

65074 

05197 

53081 

11199 

83196 

74689 

94791 

25 

36 

65690 

05926 

53958 

12273 

84542 

76424 

97111 

24 

37 

66307 

06657 

54837 

13350 

85893 

78167 

99444 

23 

38 

66925 

0739C 

55720 

14432 

87250 

79918 

9.01788 

22 

39 

67545 

08125 

56605 

15517 

88612 

81677 

04146 

21 

40 

68167 

08863 

57493 

16607 

89979 

83443 

06515 

20 

41 

4.68791 

5.09602 

5.58383 

6.17700  6 

.91352 

7.85218 

9.08897 

19 

42 

69417 

10344 

59277 

18797; 

92731 

87001 

11292 

18 

43 

70044 

11088 

60174 

19898. 

94115 

88792 

13C99 

17 

44 

70673 

11835 

61073 

21004 

95505 

90592 

16120 

16 

45 

71303 

1258? 

61976 

22113 

96900 

92400 

18553 

15 

46 

71935 

13334 

62881 

23226 

98301 

94216 

20999 

14 

47 

72569 

14087 

63790 

24343 

99708 

96040 

23459 

13 

48 

73205 

14842 

64701 

25464  7 

.01120 

97873 

25931 

12 

49 

73843 

15599 

65616 

20590 

02538 

99714 

28417 

11 

50 

74482 

16359 

66533 

27719 

03962 

8.01565 

30917 

10 

51 

4.75123 

5.17121 

5.67454 

6.28853  7 

.05392 

8.03423 

9.33430 

9 

52 

75T66 

17886 

68377 

29991 

06828 

05291 

35957 

8 

53 

76411 

18652 

69304 

31133 

08269 

07167 

38497 

54 

77057 

19421 

70234 

32279 

09717 

09052 

41052 

6 

55 

77705 

20193 

71166 

33429 

1117f 

10946 

43620 

5 

56 

78355 

20906 

72102 

34584 

12630 

12849 

46203 

4 

57 

79007 

21742 

73041 

35743 

14096 

14760 

48800 

3 

58 

79661 

22521 

73983 

36906 

15568 

16681 

51411 

2 

59 

80316 

23301 

74929 

38073 

17046 

18612 

54037 

1 

60 

80973 

24084 

75877 

39245 

18530 

20551 

56677 

0 

/ 

12° 

11° 

10° 

9° 

8° 

7° 

6° 

/ 

COSECANTS. 

NATURAL  SECANTS  AND   COSECANTS         275 


SECANTS. 

/ 

84° 

85° 

86° 

87° 

88° 

89° 

o 

9.56677 

11.47371 

14.33559 

19.10732 

28.65371 

57.298G9 

60 

1 

59332 

51199 

39547 

21397 

89440 

58.26976 

59 

2 

62002 

55052 

45586 

32182 

29.13917 

59.27431 

58 

3 

64687 

58932 

51676 

43088 

38812 

60.31411 

57 

4 

67387 

62837 

57817 

541  19 

64137 

61.39105 

56 

5 

70103 

66769 

64011 

65275 

89903 

62.50715 

55 

6 

72833 

70728 

70258 

76560 

30.16120 

63.66460 

54 

7 

75579 

74714 

76558 

87976 

42802 

64.86572 

53 

8 

78341 

78727 

82913 

99524 

69960 

66.11304 

52 

9 

81119 

82768 

89323 

20.11208 

97607 

67.40927 

51 

10 

83912 

86837 

95788 

23028 

31.25758 

68.75736 

50 

11 

9.86722 

11.90934 

15.02310 

20.34989 

31.54425 

70.16047 

49 

12 

89547 

95060 

08890 

47093 

83623 

71.62207 

48 

13 

92389 

99214 

15527 

59341 

32.13366 

73.14583 

47 

14 

95248 

12.03397 

22223 

71737 

43671 

74.73586 

46 

15 

98123 

07610 

28979 

84283 

74554 

76.39655 

45 

16 

10.01015 

11852 

35795 

96982 

33.06030 

78.13274 

44 

17 

03923 

16125 

42672 

21.09838 

38118 

79.94968 

43 

18 

06849 

20427 

49611 

22852 

70835 

81.85315 

42 

19 

09792 

24761 

56614 

36027 

34.04199 

83.84947 

41 

20 

12752 

29125 

63679 

49368 

38232 

85.94561 

40 

21 

10.15730 

12.33521 

15.70810 

21.02876 

34.72952 

88.14924 

39 

22 

18725 

37948 

78005 

76555 

35.08380 

90.46886 

38 

23 

21739 

42408 

85268 

90409 

44539 

92.91387 

37 

24 

24770 

46900 

92597 

22.04440 

81452 

95.49471 

36 

25 

27819 

51424 

99995 

18653 

36.19141 

98.22303 

35 

26 

30887 

55982 

16.07462 

33050 

57633 

101.11185 

34 

27 

33973 

60572 

14999 

47635 

96953 

104.17574 

33 

28 

37077 

65  197 

22607 

62413 

37.37127 

107.43114 

32 

29 

40201 

69856 

30287 

77386 

78185 

110.89656 

31 

30 

43343 

74550 

38041 

92559 

38.20155 

114.59301 

30 

31 

10.46505 

12.79278 

16.45869 

23.07935 

38.63068 

118.54440 

29 

32 

49685 

84042 

53772 

23520 

39.06957 

122.77808 

28 

33 

52886 

88841 

61751 

39316 

51855 

127.32526 

27 

34 

56106 

93677 

69808 

553-J9 

97797 

132.22229 

26 

35 

59346 

98549 

77944 

71563 

40.44820 

137.51108 

25 

36 

62605 

13.03158 

86159 

88022 

92963 

143.24061 

24 

37 

65885 

08404 

94456 

24.04712 

41.42266 

149.46837 

23 

38 

69186 

13388 

17.02835 

21637 

92772 

156.26228 

22 

39 

72507 

18411 

11297 

38802 

42.44525 

163.70325 

21 

40 

75849 

23472 

19843 

56212 

97571 

171.88831 

20 

41 

10.79212 

13.28572 

17.28476 

24.73873 

43.51961 

180.93496 

19 

42 

82596 

33712 

37196 

91790 

44.07746 

190.98680 

18 

43 

86001 

38891 

46005 

25.09969 

64980 

202.22122 

17 

44 

89428 

44112 

54903 

28414 

45.23720 

214.85995 

16 

45 

92877 

49373 

63893 

47134 

84026 

229.18385 

15 

46 

96348 

54676 

72975 

66132 

46.45963 

245.55402 

14 

47 

99841 

60021 

82152 

85417 

47.09596 

264.44269 

13 

48 

11.03356 

65408 

91424 

26.04994 

74997 

286.47948 

12 

49 

06894 

70838 

18.00794 

24869 

48.42241 

312.52297 

11 

50 

10455 

76312 

10262 

45051 

49.11406 

343.77516 

10 

51 

11.14039 

13.81829 

18.19830 

26.65546 

49.82576 

381.97230 

9 

52 

17646 

87391 

29501 

863GO 

50.55840 

,429.71873 

8 

53 

21277 

92999 

39274 

27.07503 

51.31290 

491.10702 

7 

54 

24932 

98651 

49153 

28981 

52.09027 

572.95809 

6 

55 

28610 

14.04350 

59139 

50804 

89156 

687.54960 

5 

56 

32313 

10096 

69233 

72978 

53.71790 

859.43689 

4 

57 

36040 

15889 

79438 

95513 

54.57046 

1145.9157 

3 

58 

39792 

21730 

89755 

28.18417 

55.45053 

1718.8735 

2 

59 

43569 

27620 

19.00185 

41700 

56.35946 

3437.7468 

1 

60 

47371 

33559 

10732 

65371 

57.29869 

oo 

0 

i 

6° 

4° 

3° 

2° 

1° 

0° 

/ 

COSECANTS. 

276    NATURAL  VERSED   SINES   AND  EXTERNAL  SEC. 
0°  1°  2°  3° 


< 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

/ 

0 

1 

2 
3 
4 

00000 
00000 
00000 
.00000 
.  00000 

.00000 
.00000 
.00000 
.00000 
.00000 

•00015 
-00016 
-00016 
.00017 
.00017 

.00015 
.00016 
.00016 
.00017 
-00017 

•00061 
.00062 
•00063 
•00064 
-00065 

.00061 
.00062 
.00063 
.00064 
-00065 

00137 
00139 
00140 
00142 
•00143 

•00137 
•00139 
.00140 
.00142 
-00143 

0 

1 
2 
3 
4 

5 
6 
7 
6 
9 

•00000 
.00000 
.00000 
-00000 
.00000 

.00000 
.00000 
.00000 
.00000 
.00000 

-00018 
.00018 
-00019 
.00020 
.00020 

•00018 
.00018 
.00019 
.00020 
-00020 

-00066 
-00067 
-00068 
-00069 
-00070 

.00066 
.00067 
.00068 
•00069 
-00070 

•00145 
.00146 
-00148 
.00150 
.00151 

-00145 
.00147 
•00148 
.00150 
-00151 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

.  00000 
.00001 
.00001 
•00001 
.00001 

.00000 
.00001 
.00001 
.00001 
.00001 

-00021 
-00021 
.00022 
-00023 
•00023 

.00021 
.00021 
.00022 
.00023 
.00023 

-00071 
-00073 
.  00074  " 
.00075 
-00076 

.00072 
.00073 
-00074 
•00075 
.00076 

.00153 
•00154 
.00156 
-00158 
-00159 

.00153 
•00155 
.00156 
•00158 
•00159 

10 

11 

12 
13 
14 

15 
16 
17 
18 
19 

.00001 
•00001 
.00001 
•00001 

.00002 

.00001 
.00001 
.00001 
.00001 
.00002 

•00024 
.00024 
•00025 
.00026 
.00026 

.00024 
.00024 
.00025 
.00026 

-00026 

.00077 
.00078 
.00079 
-00081 

.00082 

.00077 
-00078 
-00079 
.00081 
.00082 

.00161 
•00162 
•00164 
•00166 
.00168 

•00161 
•00163 
.00164 
.00166 
-00168 

15 
18 
17 
18 
19 

20 

21 
22 
23 
24 

.00002 
.00002 
00002 
.00002 
00002 

.00002 
.00002 
-00002 
.00002 
-00002 

-00027 
•00028 
-00028 
.00029 
.00030 

-00027 
.00028 
-00028 
.00029 
-00030 

-00083 
-00084 
00085 
.00087 
-00088 

.00083 
•00084 
-00085 
-00087 
-00088 

.00169 
-00171 
-00173 
-00174 
.00176 

-00169 
•00171 
•00173 
•00175 
.00176 

20 

21 
22 
23 
24. 

25 
26 
27 
28 
29 

.00003 
00003 
.  00003 
.00003 
.  00004 

.  00003 
.00003 
.00003 
.00003 
.00004 

-00031 
-00031 
.00032 
-00033 
00034 

•OOC3  1 
00031 
.00032 
.00033 
-00034 

•00089 
00090 
-00091 
.00093 
.00094 

-00089 
•00090 
•00091 
-00093 
-00094 

-0017^ 
•00179 
•00181 
.00183 
-00185 

•00178 
.00180 
•00182 
•00183 
-00185 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

.  00004 
.00004 
.  00004 
.00005 
-00005 

.00004 
.  00004 
.  00004 
•00005 
.00005 

00034 
•00035 
.00036 
.00037 
.00037 

.00034 
.00035 
•00036 
.00037 
-00037 

00095 
.00096 
.00098 
•00099 
•00100 

-OOOS5 
.00097 
.00098 
.00099 
.00100 

00187 
00188 
•00190 
•00192 
•00194 

•00187 
•00189 
-00190 
•00192 
•00194 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

00005 
.00005 
.00006 
.00006 
.00006 

.00005 
.00005 
.00006 
00006 
.00006 

.00038 
.00039 
.00040 
•00041 

.00041 

-00038 
.00039 
•00040 
•00041 
.00041 

•00102 
•00103 
.00104 
•00106 
.00107 

.00102 
•00103 
•00104 
.00106 
.00107 

•00196 
•00197 
.00199 
-00201 

.00203 

•00196 
-00198 
-00200 
•00201 
•00203 

35 
36 
37 
38 
_39 

40 

41 
42 
43 
44 

•00007 
00007 
.00007 
00008 
00008 

.00007 
•00007 
.00007 
.00008 
00008 

-00042 
•00043 
00044 
•00045 
•00046 

.00042 
•00043 
-  00044 
.00045 
-00046 

-00108 
•00110 
•00111 
-00112 
-00114 

•00108 
.00110 
-00111 
-00113 
00114 

.00205 
•00207 
•00208 
.00210 
00212 

•00205 
•00207 
•00209 
•00211 
-00213 

40 

41 
42 
43 
44 

45 
46 
47 
47 
49 

00009 
00009 
.00009 
.00010 
00010 

00009 
.00009 
.00009 
•00010 
•00010 

•00047 
00048 
•00048 
.00049 
.00050 

-00047 
-00048 
-00048 
-00049 
•00050 

-00115 
•  OOU7 
-00118 
•00119 
.00121 

•00115 
00117 
•00118 
-00120 
-00121 

•00214 
•00216 
•00218 
.00220 
-00222 

00215 
.'0216 
-00218 
-00220 
•00222 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

.00011 
.00011 
.00011 
00012 
.00012 

.00011 
•00011 
.00011 
•00012 
.00012 

•00051 
•00052 
•00053 
.00054 
.00055 

•00051 
.00052 
•00053 
.00054 
•00055 

-00122 
•00124 
•00125 
.00127 
.00128 

•00122 
•00124 
.00125 
•00127 
•00128 

.00224 
•00226 
.00228 
00230 
•00232 

•00224 
•00226 
•00228 
•00230 
•00232 

50 

51 

52 
53 
54 

55 
56 
57 
58 
59 

•00013 
•00013 
00014 
•00014 
•00015 

•00013 
.00013 
-00014 
.00014 
.00015 

00056 
.00057 
.00058 
•00059 
•00060 

.00056 
•00057 
•00058 
•QUO  59 
00060 

00130 
-00131 
00133 
•00134 
.00136 

•00130 
•00131 
00133 
•00134 
•00136 

•00234 
•00236 
00238 
•00240 
.00242 

-00234 
•00236 
•00238 
.00240 
.  00242 

55 
56 
57 
58 
59 

60 

.00015 

•00015 

-00061 

•00061 

00137 

-00137 

•00244 

•00244 

60 

NATURAL  VERSED   SINES   AND   EXTERNAL  SEC.    277 
4°  5°  6°  7° 


t 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

' 

o 

1 

2 
3 
4 

.00244 
00246 
.00248 
.00250 
-00252 

•00244 
•00246 
.00248 
.00250 
-00252 

-00381 
00383 
.00386 
.00388 
.00391 

•00382 
•00385 
.00387 
-00390 
-00392 

•00548 
.00551 
-00554 
-00557 
.00560 

•00551 
•00554 
•00557 
•00560 
•00563 

-00745 
-00749 
00752 
00756 
.00760 

.00751 
•00755 
•00758 
•00762 
00765 

0 

] 
2 
3 
4 

5 
6 
7 
8 
9 

.00254 
.00256 
-00258 
•00260 
.00262 

•00254 
•00257 
•00259 
•00261 
00263 

.00393 
-00396 
.00398 
•00401 
-00404 

•00395 
.00397 
•00400 
•00403 
•00405 

•00563 
•00566 
•00569 
•00572 
•00576 

•00566 
•00569 
•00573 
.00576 
-00579 

00763 
•00767 
00770 
-00774 
-00778 

•00769 
00773 
•00776 
•00780 
.00734 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

00264 
.00266 
.00269 
•00271 
00273 

•00265 
.00267 
.00269 
•00271 
.00274 

-00406 
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-00412 
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•00417 

.  00408 
.00411 
•00413 
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00579 
•00582 
00585 
00588 
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•00582 
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•00588 
•00592 
•00595 

•00781 
•00785 
-00789 
00792 
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•00787 
•00791 
•00795 
•00799 
•00802 

10 

11 
12 
13 
14 

15 
16 
17 
18 
.19 

.00275 
-00277 
•00279 
.00281 

.00284 

.00276 
.00278 
-00280 
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•00284 

-00420 
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•00428 
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•00421 
•00424 
•00427 
•00429 

.00432 

00594 
•00598 
-00601 
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•00598 
•00601 
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-00608 

.00611 

-00800 
-  00803 
-00807 
-00811 
•00814 

-00806 
•00810 
.00813 
•00817 
•00821 

15 
16 
17 
18 
19 

2O 

21 
22 
23 
24 

•00286 
00288 
-00290 
-00293 
-00295 

•00287 
•00289 
•00291 
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.00433 
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.  00443 
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-OC610 
-00614 
•00617 
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00614 
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•00627 

-00818 
•00822 
•00825 
•00829 
.00833 

-00825 
.00828 
•00832 
.00836 
.  00840 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

-00297 
-00299 
-00301 
-00304 
.00306 

.00298 
•00300 
•00302 
•00305 
•00307 

•  00447 
•  00449 
•00452 
•00455 
•00458 

.00449 
.00451 
.00454 
•00457 
•00460 

•00626 
•00630 
•00633 
•00636 
•  00640 

•00630 
•00634 
•00637 
•00640 
•00644 

-00837 
.00840 
.00844 
-00848 
.00852 

.00844 
-00848 
-00851 
.00855 
•00859 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

.00308 
-00311 
.00313 
00315 
•00317 

•00309 
•00312 
.00314 
.00316 
•00318 

•00460 
-00463 
•00466 
•00469 
•00472 

•00463 
•00465 
•00468 
•00471 
-00474 

•  00643 
•00646 
•00649 
•00653 
.00656 

•00647 
•00650 
-00654 
•00657 
•00660 

-00856 
00859 
-00863 
-00867 
-00871 

•00863 
•00867 
.00871 
•00875 
•00878 

30 

31 
32 
33 

34 

35 
36 
37 
38 
39 

•00320 
•00322 
•00324 
.00327 
.00329 

•00321 
•00323 
•00326 
•00328 
.00330 

.  00474 
.00477 
.  00480 
.00483 
•00486 

•00477 
•00480 
•00482 
•00485 

.00483 

•00659 
•00663 
•00666 
•00669 

.00673 

•00664 
•00667 
•00671 
•00674 

.00677 

-00875 
•OC878 
•00882 
.00886 
.00890 

.00882 
00888 
.00890 
•00894 
.00898 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

.00332 
•00334 
•00336 
-00339 
-00341 

•00333 
•00335 
•00337 
•00340 
•00342 

-00489 
.00492 
-  00494 
-00497 
-00500 

-00491 
.00494 
•00497 
•00500 
•00503 

.00676 
•00680 
•00683 
.00686 
-00690 

.00681 
.00684 
•00688 
•00691 
•00695 

•00894 
•00898 
•00902 
•00906 
•00909 

.00902 
.00906 
.00910 
.00914 
•00918 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

•00343 
-00346 
-00348 
00351 
00353 

•00345 
•00347 
00350 
•00352 
•00354 

-00503 
-00506 
.00509 
-00512 
.00515 

•00506 
•00509 
•00512 
.00515 
•00518 

•00693 
00607 
-00700 
•00703 
-00707 

.00698 
•00701 
•00705 
•00708 
.00712 

•00913 
00917 
00921 
00925 
•00929 

.00922 
.00926 
.00930 
•00934 
00938 

45 
46 
47 
48 
49 

50 

5i 
52 
53 
54 

•00356 
•00358 
00361 
•  0036cJ 
-00365 

•00357 
•00359 
•00362 
•00364 
-00367 

-00518 
.00521 
•00524 
•00527 
-00530 

.00521 
•00524 
•00527 
•00530 
•00533 

•00710 
.00714 
.00717 
•00721 
•00724 

•00715 
•00719 
•00722 
•00726 
•00730 

•00933 
00937 
•00941 
•00945 
•  00949 

.00942 
.00946 
.00950 
•00954 
•00958 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59_ 

.00368 
-00370 
-00373 
•00375 
.00378 

•00369 
•00372 
•00374 
00377 
•00379 

•00533 
-00536 
•00539 
.00542 
00545 

•00536 
•00539 
•00542 
•00545 
00548 

•00728 
•00731 
•00735 
•00738 
•00742 

.00733 
•00737 
•00740 
•00744 
-00747 

•00953 
•00957 
•00961 
.00965 
•00969 

.00962 
•00968 
•00970 
•00975 
.00979 

55 
56 
57 
58 
59 

60 

•00381 

•00382 

00548 

•00551 

•00745 

•00751 

•00973 

•00983 

60 

NATURAL  VERSED   SINES  AND   EXTERNAL   SEC. 

8°  9°  10°  11° 


• 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

' 

0 

2 
3 
4 

00973 
00977 
00981 
00985 
00989 

.00983 
.00987 
•00991 
•00995 
.00999 

01231 
01236 
01240 
01245 
01249 

•01247 
-01251 
.01256 
.01261 
.01265 

01519 
01524 
01529 
01534 
01540 

•01543 
.01548 
.01553 
.01558 
•01564 

01837 
01843 
01848 
01854 
01860 

•01872 
•01877 
•01883 
.01889 
•01895 

U 
1 

2 
3 

4 

5 
6 
7 
8 
9 

00994 
00998 
01002 
01006 
01010 

•01004 
•01008 
•01012 
•01016 
.01020 

01254 
01259 
01263 
01268 
01272 

.01270 
•01275 
.01279 
•01284 
•01289 

01545 
01550 
01555 
01560 
01565 

.01569 
.01574 
.01579 
.01585 
.01590 

01865 
01871 
01876 
01382 
01888 

.01901 
.01906 
•01912 
•01918 
•01924 

5 
6 
7 
8 
9 

tf 

12 
13 
H 

01014 
01018 
01022 
01027 
01031 

.01024 
•01029 
-01033 
.01037 
.01041 

01277 
01282 
01286 
01291 
01296 

.01294 
.01298 
.01303 
•01308 
•01313 

01570 
01575 
.01580 
.01586 
.01591 

.01595 
.01601 
.01606 
.01611 
•01616 

•01893 

•01899 
•01904 
•01910 
•01916 

-01930 
•01936 
•01941 
•01947 
•01953 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

01035 
01039 
01043 
01047 
01052 

.01046 
.01050 
.01054 
.01059 
•01063 

01300 
01305 
01310 
01314 
01319 

.01318 
•01322 
.01327 
•01332 
.01337 

.01596 
01601 
.01606 
•01612 
.01617 

.01622 
.01627 
.01633 
.01638 
.01643 

•01921 
•01927 
•01933 
•01939 
-01944 

•01959 
-01965 
-01971 
.01977 
.01983 

15 
16 
17 
18 
-12 

20 

21 
22 
23 
24 

01056 
01060 
01064 
01069 
01073 

.01067 
.01071 
.01076 
.01080 
.01084 

01324 
01329 
01333 
01338 
01343 

•01342 
•01346 
•01351 
.01356 
•01361 

•01622 
.01627 
.01632 
•01638 
.01643 

•01649 
•01651 
.01659 
•01665 
.01670 

-01950 
•01956 
•01961 
•01967 
.01973 

•01989 
.01995 
-02001 
-02007 
•02013 

30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

.01077 
.01081 
.01086 
.01090 
.01094 

.01089 
.01093 
.01097 
•01102 
.01106 

01348 
01352 
.01357 
.01362 
.01367 

•01366 
.01371 
-01376 
•01381 
•01386 

•01648 
•01653 
•01659 
•01664 
.01669 

.01676 
.01681 
•01687 
.01692 
.01698 

-01979 
.01984 
•01990 
•01996 
•02002 

•02019 
•02025 
•02031 
•02037 
•02043 

25 
28 
27 
28 

29 

30 

31 
32 
33 
34 

.01098 
.01103 
.01107 
•01111 
.01116 

.01111 
.01115 
.01119 
.01124 
.01128 

.01371 
.01376 
.01381 
-01386 
•01391 

•01391 
•01395 
.01400 
-01405 
-01410 

-01675 
.01680 
•01685 
-01690 
-01696 

.01703 
.01709 
.01714 
.01720 
.01725 

•02008 
•02013 
•02019 
•02025 
•02031 

.02049 
.02055 
.02061 
.02067 
.02073 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

.01120 
.01124 
•01129 
-01133 
•01137 

.01133 
.01137 
.01142 
.01146 
-01151 

•01396 
•01400 
.01405 
.01410 
.01415 

.01415 
.01420 
•01425 
.01430 

•01435 

-01701 
-01706 
-01712 
01717 

-01723 

.01731 
.01736 
.01742 
•01747 
•01753 

•02037 
•02042 
.02048 
•02054 
-02060 

•02079 
•02085 
•02091 
.02097 
-02103 

35 
36 
37 
38 
-39 

40 

41 
42 
43 
44 

01142 
•01146 
•01151 
-01155 
•01159 

.01155 
.01160 
•01164 
•01169 
01173 

-01420 
•01425 
•01430 
01435 
.01439 

.01440 
.01445 
01450 
-01455 
•01461 

-01728 
01733 
-01739 
-01744 
-01750 

•01758 
.01764 
.01769 
.01775 
.01781 

-02066 
•02072 
•02078 
•02084 
•02090 

.02110 
.02116 
.02122 
.02128 
.02134 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

•01164 
-01168 
01173 
.01177 
-01182 

•01178 
•01182 
•01187 
.01191 
•01196 

•01444 
.01449 
.01454 
•01459 
.01464 

.01466 
.01471 
.01476 
.01481 
.01486 

-01755 
•01760 
01766 
•01771 
•01777 

01786 
.01792 
.01793 
.01803 
.01809 

•02095 
•02101 
•02107 
•02113 
•02119 

•02140 
•02146 
•02153 
•02159 
•02165 

45 
46 
47 
48 
49 

50 

51 
S2 
53 
64 

01188 
•01191 
-01195 
-01200 
-01204 

•01200 
•01205 
.01209 
.01214 
.01219 

•01469 
•01474 
•01479 
•01484 
.01489 

•01491 
•01496 
.01501 
•01506 
•01512 

•01782 
•01788 
•01793 
•0179U 
•01804 

.01815 
.01820 
-01826 
.01832 
01837 

•02125 
-02131 
•02137 
•02143 
•02149 

•02171 
•02178 
.02184 
.02190 
.02196 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

.01209 
•01213 
•01218 
•01222 
-01227 

.01223 
•01228 
.01233 
.01237 
.01242 

•01494 
•01499 
.01504 
.01509 
-01514 

.01517 
.01522 
•01527 
.01532 

•01537 

.01810 
.01815 
•01821 
.01826 
•01832 

•01843 
•01849 
.01854 
.01860 
•01866 

.02155 
•02161 
•02167 
•02173 

-02179 

.02203 
•02209 
.02215 
•02221 
.02228 

55 
56 
57 
58 
59 

60 

01231 

•01247 

.01519 

.01543 

•01837 

.01872 

•02185 

.02234 

60 

NATURAL  VERSED  SINES  AND  EXTERNAL  SEC/ 

13°  13°  14°  15° 


t 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

' 

o 

1 

2 
3 

.02185 
•02191 
•02197 
.02203 
-02210 

.02234 
.02240 
.02247 
.02253 
•02259 

.02563 
•02570 
•02576 
.02583 
.02589 

•02630 
.02637 
.02644 
.02651 
.02658 

02970 
02977 
02985 
02992 
02999 

•03081 
.03069 
.03076 
•03084 
•03091 

•03407 
•03415 
•03422 
•03430 
•03438 

•03528 
.03536 
•03544 
•03552 
•03560 

0 

i 

5 
6 

8 
9 

.02216 
•02222 
.02228 
.02234 
.  02240 

.02266 
.02272 
.02279 
.02285 
•02291 

.02596 
.02602 
•02609 
•02616 
•02622 

.02665 
•02672 
.02679 
•02686 
•02693 

03006 
03013 
03020 
•03027 
•03034 

.03099 
.03106 
•03114 
•03121 
•03129 

•03445 
•03453 
03460 
•03468 
.03476 

•03568 

.03576 
•03584 
•03592 
•03601 

5 

1 

8 

9 

10 

11 
12 
13 
14 

•02246 
•02252 
•02258 
•02265 
•02271 

•02298 

02304 
.02311 
•02317 
•02323 

•02629 
•02635 
•02642 
•02649 
.02655 

•02700 

•02707 
-02714 
.02721 
.02728 

•03041 

•03048 
•03055 
•03063 
•03070 

•03137 
.03144 
•03152 
•03159 
•03167 

•03483 
.03491 
•03498 
.03506 
.03514 

•03609 
•03617 
•03625 
•03633 
•03642 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

•  02277 
•02283 
•02289 
•02295 
•02302 

.02330 
•02336 
•02343 
.02349 
.02356 

.02662 
•02669 
•02675 
•02682 
.02689 

•02735 
•02742 
•02749 
.02756 
•02763 

.03077 
.03084 
•03091 
•03098 
•03106 

•03175 
.03182 
.03190 
•03198 
•03205 

•03521 

•03529 
.03537 
•03544 
•03552 

•03650 
•03658 
•03666 
•03674 
•03683 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

•02308 
•02314 
•02320 
•02327 
.02333 

•02362 
•02369 
•02375 
•02382 
•02388 

•02696 
•02702 
•02709 
•02716 
•02722 

.02770 
.02777 
.02784 
.02791 
.02799 

.03113 
•03120 
•03127 
•031S4 
.03142 

•03213 
•03221 
•03228 
•03236 
•03244 

•03560 
•03567 
•03575 
•03583 
.03590 

•03691 
.03699 
•03708 
•03716 
•03724 

30 

21 

22 
23 
24 

25 
26 
27 
28 
29 

•02339 
•02345 
•02352 
.02358 
.02364 

•02395 
•02402 
•02408 
•02415 
.02421 

•02729 
•02736 
•02743 
•02749 
•02756 

.02806 
•02813 
.02820 
•02827 
.02834 

•03149 
.03156 
•03163 
03171 
•03178 

.03251 
.03259 
•03267 
•03275 
•03282 

.03598 

-03606 
•03614 
.03621 
.03629 

•03732 
•03741 
•03749 
•03758 
•03766 

25 
26 
27 
23 
29 

30 

31 
32 
33 
34 

•02370 
02377 
•02383 
•02389 
•02396 

•02428 
.02435 
•02441 
.02448 
.02454 

•02763 
•02770 
•02777 
•02783 
.02790 

.02842 
•02849 
.02856 
.02863 
•02870 

•03185 
•03193 
•03200 
•03207 
•03214 

•03290 
•03298 
•03306 
•03313 
•03321 

•03637 
.03645 
.03653 
•03660 
•03668 

•03774 
•03783 
•03791 
•03799 
•03808 

30 

31 
32 
33 
34 

3o 
36 
37 
38 
,39 

•  02402 
02408 
•02415 
02421 

•02427 

•02461 
•02468 
•02474 
•02481 
.02488 

•02797 
•02804 
•02811 
•02818 
•02824 

.02878 
.02885 
.02892 
•02899 
•02907 

•03222 
•03229 
•03236 
•03244 
•03251 

•03329 
•03337 
•03345 
•03353 
03360 

•03676 
•03684 
•03692 
•03699 
•03707 

•03816 
03825 
•03833 
-03842 
•03850 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

.02434 
•  02440 
•02447 
•02453 
•02459 

•02494 
•02501 
•02508 
•02515 
.02521 

•02831 
•02838 
•02845 
02852 
•02859 

•02914 
•02921 
.02928 
•02936 
.02943 

•03258 
•03266 
•03273 
•03281 
•03288 

•03368 
•03376 
-03384 
•03392 
•  03400 

03715 
•03723 
•03731 
.03739 
•03747 

•03858 
.03867 
•03875 
.03884 
•03892 

40 

41 

42 
43 
44 

45 
46 
47 
48 
49 

•02466 
.02472 
•02479 
.02485 
•02492 

•02528 
•02535 
•02542 
.02548 
•02555 

•02866 
•02873 
•02880 
•02887 
-02894 

.02950 
.02958 
•02965 
.02972 
.02980 

•03295 
•03303 
•03310 
•03318 
.03325 

•03408 
•03416 
.03424 
•03432 
•03439 

•03754 
•03762 
03770 
•03778 
•03786 

•03901 
•03909 
•03918 
•03927 
.03935 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

.02498 
•02504 
•02511 
•02517 
02524 

•02562 
•02569 
•02576 
•02582 
•02589 

.02900 
•02907 
.02914 
•02921 
.02928 

•02987 
•02994 
•03002 
.03009 
•03017 

03333 
•03340 
•03347 
•03355 
•03362 

•03447 
•03455 
•03463 
•03471 
•03479 

•03794 
•03802 
•03810 
•03818 
•03826 

.03944 
.03952 
•03961 
.03969 
.03978 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

02530 
•02537 
•02543 
•02550 
•02556 

•02596 
•02603 
•02610 
•02617 

.02624 

•02935 

.02942 
02949 
•02956 
•02963 

.03024 
.03032 
.03039 
03046 
•03054 

•03370 
•03377 
03385 
•03392 
•03400 

•03487 
•03495 
•03503 
•03512 
•03520 

.03834 
.03842 
•03850 
•03858 
•03866 

•03987 
.03995 
•04004 
•04013 
•  04021 

55 
56 
57 
58 
59 

60 

-02563 

•02630 

•02970 

.03061 

03407 

•03528 

•03874 

.04030 

60 

280    NATURAL  VERSED  SINES   AND   EXTERNAL  SEC. 

16°  17°  18°  19° 


' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec 

t 

0 

1 

2 
3 
4 

•03874 
-03882 
-03890 
03898 
-03906 

.04030 
.04039 
.04047 
•04056 
.04065 

•04370 
•04378 
•04387 
•04395 
.  04404 

•04569 
.04578 
•04588 
.04597 
.04606 

.04894 
•04903 
•04912 
.04921 
•04930 

.05146 
.05156 
.05166 
•05176 
•05186 

05448 
.05458 
•05467 
.05477 
•05486 

•05762 
05773 
•05783 
•05794 
.05805 

0 

1 

2 
3 
4 

5 
6 
7 
8 
9 

-03914 
-03922 
.03930 
.03938 
.03946 

.04073 
.04082 
•04091 
.04100 
•04108 

.04412 
•04421 
.04429 
•04438 
-04446 

•04616 
•04625 
04635 
.  04644 
.04653 

.04939 
•04948 
•04957 
•04967 
.04976 

•05196 
.05206 
.05216 
•05226 
.05236 

•05496 
•05505 
.05515 
•05524 
.05534 

•05815 
•05826 
•05836 
•05847 
-05858 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

.03954 
.03963 
.03971 
.03979 
-03987 

.04117 
04126 
.04135 
.04144 
.04152 

•04455 
.04464 
-04472 
.04481 
.  04489 

.04663 
•04672 
.04682 
.04691 
.  04700 

•04985 
.04904 
.05003 
.05012 
•05021 

.05246 
.05256 
.05266 
.05276 
.05286 

.05543 
.05553 
•05562 
.05572 
.05582 

•05869 
•05879 
•05890 
•05901 
•05911 

10 

11 
12 
13 
14 

15 

16 
17 
18 
19 

.03995 
-  04003 
-04011 
-04019 

•04028 

.04161 
.04170 
.04179 
.04188 

-04197 

•04498 
•04507 
•04515 
-04524 
-04533 

.04710 
.04719 
.04729 
.04738 
.04748 

•05030 
•05039 
•05048 
•05057 
.05067 

.05297 
.05307 
.05317 
.05327 
.05337 

•05591 
.05601 
05610 
•05620 
.05630 

.05922 
.05933 
•05944 
•05955 
.05965 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

-04036 
-  04044 
•04052 
04060 
.04069 

-04206 
.04214 
.04223 
.04232 
.04241 

-04541 
-04550 
-04559 
-04567 
.04576 

.04757 
.04767 
.04776 
•04786 
.04795 

.05076 
.05085 
•05094 
•05103 
•05112 

.05347 
-05357 
.05367 
.05378 
.05388 

.05639 
•05649 
.05658 
•05668 
•05678 

•05976 
-05987 
•05998 
•06009 
.06020 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

-04077 
-04085 
•04093 
04102 
.04110 

.04250 
•04259 
.04268 
•04277 
.04286 

.04585 
.04593 
.04602 
.04611 
.04620 

.04805 
-04815 
.04824 
04834 
.  04843 

•05122 
.05131 
•05140 
•05149 
•05158 

•05398 
.05408 
.05418 
.05429 
.05439 

.05687 
•05697 
05707 
•05716 
•05726 

•06030 
•06041 
-06052 
.06063 
.06074 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

.04118 
04126 
-04135 
04143 
-04151 

.04295 
.04304 
.04313 
•04322 
.04331 

.04628 
-04637 
-04646 
04655 
.04663 

.04853 
•04863 
•04872 
•04882 
.04891 

•05168 
.05177 
•05186 
•05195 
.05205 

•05449 
.05460 
.05470 
.05480 
.05490 

•05736 
•05746 
•05755 
.05765 
.05775 

.06085 
•06096 
.06107 
•06118 
.06129 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

-04159 
-04168 
-04176 
•04184 

.04193 

.  04340 
•  04349 
•04358 
•04367 
•04376 

.04672 
•04681 
•04690 
.04699 
.04707 

.04901 
•04911 
04920 
•04930 
-04940 

•05214 
.05223 
•05232 
•05242 
•05251 

.05501 
-05511 
.05521 
.05532 

.05542 

.05785 
.05794 
.05804 
.05814 
-05824 

•06140 
•06151 
.06162 
.06173 
•06184 

35 
36 
37 
38 
-39 

40 

41 

42 
43 
44 

.04201 
.04209 
-04218 
-04226 
-04234 

.04385 
.04394 
.04403 
.04413 
.04422 

.04716 
•04725 
-04734 
-04743 
•04752 

.04950 
•04959 
•04969 
•04979 
.04989 

•05260 
.05270 
•05279 
•05288 
.05298 

.05552 
.05563 
.05573 
.05584 
.05594 

.05833 
.05843 
•05853 
.05863 
.05873 

•06195 
•06206 
.06217 
-06228 
•06239 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

-  04243 
.04251 
04260 
.04268 
.04278 

.04431 
.  04440 
.04449 
-04458 
.04468 

-04760 
-04769 
•04778 
-04787 
•04796 

.04998 
•05008 
.05018 
.05028 
.05038 

.05307 
.05316 
.05326 
.05335 
•05344 

•05604 
.05615 
.05625 
.05636 
-05646 

.05882 
-05892 
•05902 
•05912 
.05922 

•06250 
.06261 
-06272 
.06283 
-06295 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

04285 
.04293 
•04302 
-04310 
-04319 

.04477 
•04486 
.04495 
.04504 
•04514 

•04805 
•04814 
•04323 
•04832 
-04841 

•05047 
.05057 
.05067 
.05077 
.05087 

.05354 
•05363 
.05373 
•05382 
-05391 

.05657 
.05667 
.05678 
•05688 
•05699 

-05932 
-05942 
.05951 
.05961 
.05971 

-06306 
-06317 
-06328 
•06339 
-06350 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

-04327 
.04336 
-04344 
-04353 
-043RT 

•04523 
.04532 
•04541 
.04551 
.04560 

-04850 
-04858 
-04867 
•04876 
04885 

.05097 
•05107 
.05116 
.05126 
.05136 

.05401 
.05410 
.05420 
•05429 
-05439 

.05709 
.05720 
•05730 
.05741 
-05751 

•05981 
•05991 
•06001 
.06011 

.06021 

.06362 
•06373 
•06384 
•06305 
-06407 

55 
56 
57 
58 
_59 

60 

-04370 

.04569 

•04894 

.05146 

•05448 

.05762 

.06031 

.06418 

60 

NATURAL  VERSED  SINES  AND  EXTERNAL  SEC.    281 
20°  21°  33°  33° 


/ 

Vers. 

Ex,  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

/ 

0 

1 

2 
3 
4 

.06031 
-06041 
.06051 
•06061 
•06071 

•06418 
.06429 
•06440 
-06452 
06463 

-06642 
•06652 
•06663 
•06673 
.06684 

•07115 
.07126 
•07138 
.07150 
.07162 

.07282 
.07293 
•07303 
•07314 
•07325 

•07853 
•07866 
•07879 
•07892 
07904 

•07950 
•07961 
•07972 
•07984 
-07995 

.08636 
•08649 
•08663 
-08676 
•08690 

0 

1 
2 
3 

4 

5 
6 
7 
8 
9 

•06081 
.06091 
•06101 
.06111 
06121 

.06474 
.06486 
06497 
•06508 
.06520 

•06694 
•06705 
•06715 
•06726 
•06736 

.07174 
.07186 
•07199 
.07211 
.07223 

•07336 
•07347 
•07358 
-07369 
.07380 

•07917 
07930 
07943 
•07955 
.07968 

-08006 
-08018 
-08029 
-08041 
-08052 

•08703 
•08717 
•08730 
-08744 
•08757 

5 
6 

8 
9 

10 

11 
12 
13 
14 

.06131 
•06141 
•06151 
.06161 
.06171 

•06531 
.06542 
.06554 
.06565 
•06577 

.06747 
•06757 
-06768 
•06778 
-06789 

•07235 
•07247 
•07259 
•07271 
.07283 

•07391 
-07402 
•07413 
•07424 
•07435 

.07981 
•07994 
•08006 
.08019 
•08032 

-08064 
-08075 
-08086 
-08098 
•08109 

•08771 
-08784 
•08798 
-08811 
-08825 

10 

11 

12 
13 
14 

15 
16 
17 
18 
19 

•06181 
•06191 
•06201 
•06211 

.06221 

•06588 
.06600 
.06611 
.06622 
.06634 

•06799 
-06810 
-06820 
-06831 
.06841 

•07295 
•07307 
-07320 
•07332 

.07344 

•07446 
•07457 
•07468 
•07479 
-07490 

•08045 
•08058 
•08071 
-08084 
•08097 

•08121 
-08132 
.08144 
-08153 
08167 

•08839 
.08852 
-08866 
.08880 
•08893 

15 
18 
17 
18 
19 

30 

21 
22 
23 
24 

.06231 
.06241 
.06252 
•06262 
.06272 

.06645 
•06657 
.06668 
•06680 
.06691 

-06852 
-06863 
.06873 
•06884 
.06894 

•07356 
•07368 
.07380 
.07393 
.07405 

-07501 
-07512 
•07523 
•07534 
•07545 

•08109 
•08122 
•08135 
•08148 
.08161 

•08178 
-08190 
•08201 
-08213 
-OC225 

.08907 
.08921 
08934 
-08948 
.08962 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

•06282 
•06292 
•06302 
.06312 
.06323 

.06703 
.06715 
.06726 
.06738 
.06749 

•06905 
.06916 
-06926 
-06937 
-06948 

.07417 
.07429 
.07442 
.07454 
.07486 

•07556 
•07568 
-07579 
.07590 
.07601 

.08174 
•08087 
•08200 
•08213 
•08226 

-08236 
08248 
-08259 
•08271 
0*282 

.08975 
-08989 
-09003 
•09017 
.09030 

25 
28 
27 
28 
29 

30 

31 
32 
33 
34 

•06333 
•06343 
•06353 
.06363 
•06374 

•06761 
.06773 
.06784 
.06796 
.06807 

-06958 
.06969 
-06980 
•06990 
-07001 

•07479 
•07491 
•07503 
.07516 
.07528 

•  0761-2 
07623 
•07634 
•07645 
•07657 

.08239 
.08252 
•08265 
•08278 
•08291 

-08294 
-08306 
•08317 
•08329 
-08340 

•09044 
.09058 
-09072 
•09086 
.09099 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

•06384 
•06394 
•06404 
•06415 

.06425 

•06819 
.06831 
•06843 
.06854 
.06866 

.07012 
•07022 
-07033 
.07044 
•07055 

.07540 
.07553 
•07565 
.07578 
.07590 

.07668 
-07679 
-07690 
07701 

•07713 

•08305 
.08318 
.08331 
.08344 
.08357 

-08352 
-08364 
-08375 
•08387 
•08399 

•09113 
09127 
-09141 
•09155 
•09169 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

.06435 
-06445 
•06456 
•06466 
.06476 

-06878 
•06889 
06901 
.06913 
.06925 

-07065 
•07076 
•07087 
.07098 
•07108 

-07J02 
•07615 
•07627 
•07b40 
•07652 

•07724 
•07735 
•07746 
•07757 
•07769 

.08370 
.08383 
08397 
•08410 
•  08423 

-08410 

-08422 
08434 
.08445 
•08457 

-09183 
•09197 
•09211 
•09224 
•09238 

4O 

41 
42 
48 
44 

45 
46 
47 
48 
49 

06486 
.06497 
•06507 
•06517 
•06528 

.06936 
-06948 
•06960 
.06972 
06984 

•07119 
-07130 
•07141 
•07151 
•07162 

•07665 
•07677 
•07690 
•07702 
.07715 

07780 
•07791 
•07802 
07814 
•07825 

•08436 
.  08449 
•08463 
-08476 
.08489 

08469 
•08481 
•08492 
•08504 
•08510 

.09252 
•09266 
•09280 
•09294 
09308 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

•06538 
•06548 
•06559 
06569 
.06580 

•06995 
.07007 
-07019 
•07031 
.  07043 

-07173 
-07184 
.07195 
-07206 
-07216 

.07727 
•07740 
•07752 
•07765 
.07778 

•07836 
07848 
07859 
.07870 
•07881 

.08503 
.08516 
.08529 
.08542 
•08556 

•08528 
•08539 
.08551 
•08563 
.08575 

•09323 
.09337 
.09351 
•09365 
09379 

50 

51 
52 
53 
54 

55 
56 
57 
58 
-SL- 

.06590 
•06600 
06611 
.06621 
-06632 

.07055 
.07067 
.07079 
•07091 
-07103 

.07227 
.07238 
-07249 
•07260 
•07271 

•07790 
•07803 
•07816 
•07828 
•07841 

•07893 
.07904 
.07915 
•07927 
-07938 

•08569 
-08582 
-08596 
-08069 
-08623 

•08586 
08598 
08610 
.08622 
08634 

.09393 
.09407 
•09421 
-09435 
•  09449 

55 
56 
57 
58 

59 

OG 

06642 

•07115 

•07282 

•07853 

.07950 

•08636 

•08645 

•09464 

60 

282    NATURAL  VERSED   SINES  AND  EXTERNAL  SEC. 
34°  35°  36°  27° 


• 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

/ 

0 

2 
3 

4 

08645 
.08657 
.08669 
.08681 
.08693 

.09464 
.09478 
.09492 
.09506 
.09520 

.09369 
.09382 
.09394 
.09406 
-09418 

•10338 
.10353 
•10368 
•10383 
.10398 

10121 
.10133 
.10146 
.10159 
•10172 

•11260 
.11276 
•11292 
.11308 
•11323 

-10899 
-10913 
•10926 
•10939 
•10952 

.12233 
•1224& 
.12266 
•12283 
.12299 

0 

1 
2 
3 

4 

5 
6 

9 

.08705 
.08717 
.08728 
.08740 
.08752 

-09535 
.09549 
.09563 
•09577 
.09592 

•09431 
•09443 
.09455 
09468 
.09480 

•10413 
•10428 
.  10443 
•10458 
.10473 

•10184 
•10197 
•10210 
.10223 
-10236 

•11339 
11355 
•11371 
•11387 
•11403 

•10965 
•10979 
-10992 
-11005 
•11019 

•12316 
•12333 
•12349 
.12366 
.12383 

5 
6 
7 

9 

1O 

11 
12 
13 
14 

.08764 
.08776 
.08788 
•088CO 
.08812 

.09608 
.09620 
.09635 
.09649 
.09663 

.09493 
.09505 
•09517 
.09530 
.09542 

.10488 

•10503 
10518 
•10533 
.10549 

•  10248 
•10261 
•10274 
•10287 
•10300 

•11419 
.11435 
•11451 
•11467 
•11483 

.11032 
•11045 
•11058 
11072 
.11085 

.  12400 
•12416 
•12433 
•12450 
.12467 

10 

11 
12 
13 
14 

15 
16 
17 
18 

19 

.08824 
.08836 
.08848 
.08860 
-08872 

.09678 
.09692 
.09707 
.09721 
-09735 

09554 
.09567 
.09579 
.09592 
.09604 

•10564 
•10579 
•10594 
.10609 
.10625 

.10313 
-10326 
-10338 
-10351 
•10364 

-11499 
11515 
-11531 
.11547 
-11563 

-11098 
-11112 
-11125 
-11138 
-11152 

-12484 
.12501 
•12518 
•125b4 
-12551 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

.08884 
.08896 
.08908 
.08920 
.08932 

.09750 
.09764 
.09779 
.09793 
.09808 

•09617 
•09629 
.09642 
.09654 
.09666 

.10640 
•10655 
•10670 
.10686 
.10701 

•10377 
.10390 
.  10403 
.10416 
10429 

-11579 
-11595 
•11611 
•11627 
.11643 

•11165 
-11178 
-11192 
.11205 
•11218 

-12568 
.12585 
.12602 
.12619 
12636 

20 

21 
22 
23 
24 

25 
28 
27 
28 
29 

.08944 
.08956 
.08968 
-08980 
-08992 

.09822 
.09837 
.09851 
.09866 
.09880 

.09679 
•09691 
.09704 
.09716 
.09729 

.10716 
.10731 
•10747 
•10762 
.10777 

.  10442 
.10455 
•10468 
•  10481 
•  10494 

•11659 
•11675 
.11691 
.11708 
.11724 

.11232 
•11245 
•11259 
-11272 
-11285 

•12653 
.12670 
.12687 
.12704 
.12721 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

.09004 
.09016 
.09028 
.09040 
.09052 

.09895 
.09909 
.09924 
•09939 
.09953 

.09741 
•09754 
.09767 
.09779 
.09792 

.10793 
10808 
10824 
.10839 
.10854 

.10507 
.10520 
.10533 
•10546 
.10559 

.11740 
.11756 
.11772 
-11789 
•11805 

-11299 
•11312 
-11326 
-11339 
.11353 

.12738 
•12755 
.12772 
•12789 
•12807 

3O 

31 
32 
33 
34 

35 
36 
37 
38 
39 

.09064 
.09076 
.09089 
.09101 
.09113 

.09968 
.09982 
.09997 
.10012 
.10026 

.09804 
09817 
09829 
.09842 
.09854 

•10870 
.10885 
.10901 
.10916 
•10932 

.10572 
.10585 
.10598 
•10611 
.10624 

.11821 
.11838 
•11854 
•11870 
-11886 

•11366 
•11380 
-11393 
-11407 

.11420 

•12824 
.12841 
.12858 
•  U875 

•12892 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

.09125 
.09137 
•09149 
.09161 
.09174 

.10041 
.10055 
.10071 
.10085 
.10100 

•09867 
.09880 
.09892 
.09905 
.09918 

-10947 
•10963 
.10978 
.10994 
.11009 

•10637 
•10650 
.10663 
•10676 
•10689 

-11903 
•11919 
•11936 
.11952 
-11968 

-11434 
-11447 
•11461 
•11474 
•11488 

12910 
.12927 
.12944 
.12961 
.12979 

4O 

41 
42 
43 
44 

45 
46 
47 
48 
49 

.09186 
-09198 
•09210 
.09222 
•09234 

.10115 
.10130 
.10144 
.10159 
.10174 

•09930 
.09943 
•09955 
.09968 
•09981 

-11025 
.11041 
.11056 
.11072 
.11087 

.10702 
.10715 
.10728 
•10741 
•10755 

.11985 
•12001 
•12018 
•12034 
•12051 

•11501 
•11515 
-11528 
-11542 
-11555 

•12996 
•13013 
.13031 
.13048 
13065 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

•09247 
-09259 
09271 
•09283 
.09296 

.10189 
.10204 
.10218 
.10233 
.10248 

•09993 
•10006 
-10019 
•10032 
.  10044 

.11103 
•11119 
•11134 
.11150 
.11166 

•10768 
•10781 
10794 
•10807 
10820 

.12067 
-  12084 
.12100 
•12117 
•  12.133 

-11569 
-11583 
-11596 
11610 
-11623 

-13083 
-13100 
-13117 
•13135 
13152 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

•09308 
.09320 
•09332 
•09345 
•09357 

.10263 
.10278 
.10293 
.10308 
•103?3 

.10057 
-10070 
.10082 
•10095 
•10108 

.11181 
.11197 
.11213 
.11229 
.11244 

•10833 
•10847 
•10860 
•10873 
•10886 

•12150 
-12166 
.12183 
.12199 
-12216 

.11637 
.11651 
.11664 
.11678 

-11692 

.13170 
-13187 
-13205 
-13222 
.13240 

55 
56 
57 
58 
59 

60 

•09369 

.10338 

•10121 

.11260 

•10899 

•12233 

•11705 

-13257 

60 

NATURAL  VERSED  SINES   AND  EXTERNAL   SEC. 

28°  39°  30°  31° 


• 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

t 

0 

1 

2 
3 
4 

•11705 
•11719 
•11733 
.11746 
•11760 

•13257 
.13275 
.13292 
.13310 
•13327 

.12538 
•12552 
.12566 
•12580 
•12595 

•14335 
•14354 
•14372 
.14391 
.  14409 

•13397 
•13412 
-13427 
•13441 
-13456 

.  15470 
.15489 
.15509 
.15528 
.15548 

•  14283 
•14298 
•14313 
•  14328 
•  14343 

•16663 
.16684 
•16704 
•16725 
•16745 

0 
1 
2 

3 
4 

5 
6 
7 
8 
9 

•11774 
•11787 
•11801 
•11815 
.11828 

•13345 
•13362 
.13380 
•13398 
•13415 

•12609 
•12623 
•12637 
•12651 
.12665 

.  14428 
.  14446 
•14465 
.14483 
.14502 

•13470 
•13485 
•13499 
•13514 
•13529 

.15567 
•15587 
-15606 
-15626 
.15645 

•14358 
•14373 
•14388 
.  14403 
•14418 

•16766 
•16786 
.16806 
•16827 
.16848 

5 
6 
7 
8 

9 

10 

11 
12 
13 
14 

•11842 
.11856 
•11870 
•11883 
.11897 

-13433 
•13451 
•13468 
.13486 
•13504 

•12679 
.12694 
.12708 
•12722 
•12736 

•14521 
•14539 
•14558 
.14576 
•14595 

.13543 
•13558 
•13573 
•13587 
-13602 

•15665 
•15684 
.15704 
.15724 
.15743 

•  14433 
•  14449 
.  14464 
.  14479 
.  14494 

.16868 
•16889 
•16909 
•16930 
.16950 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

.11911 
•11925 
•11938 
.11952 
•11966 

•13521 
.13539 
•13557 
•13575 

.13593 

.12750 
•12765 
•12779 
•12793 
•12807 

•14614 
•14632 
.14651 
.14670 
•14689 

•13616 
-13631 
.13646 
•13660 
•  13675 

•15763 
.15782 
.15802 
.15822 

.15841 

.14509 
.14524 
.14539 
•14554 
•14569 

•16971 
•16992 
•17012 
•17033 

.17054 

15 
16 
17 
18 
,19 

20 

21 
22 
23 
24 

.11980 
.11994 
.12007 
.12021 
.12035 

•13610 
.13628 
.13646 
.13664 
.13682 

•12822 
.12836 
.12850 
•12864 
•12879 

.14707 
.14726 
.14745 
•14764 
•  14782 

•13690 
•13705 
•13719 
•13734 
•13749 

.15861 
-  -15881 
.15901 
.15920 
.15940 

.14584 
•14599 
•14615 
.  14630 
•14645  . 

•17075 
•17095 
.17116 
•17137 
.17158 

•40 

21 
22 
23 
24 

25 
26 
27 
28 
29 

-12049 
.12063 
•12077 
•12091 
.12104 

•13700 
.13718 
.13735 
.13753 
.13771 

•12893 
.12907 
•12921 
•12936 
.12950 

•14801 
.14820 
•14839 
.14858 
.14877 

•13763 
•13778 
•13793 
•13808 
•13822 

.15960 
.15980 
.16000 
.16019 
-16039 

•14660 
•14675 
-14690 
.14706 
.14721 

.17178 
.17199 
.17220 
•17241 
.17262 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

•12118 
•12132 
•12146 
12160 
12174 

.13789 
.13807 
.13825 
.13843 
•13861 

•12964 
•12979 
.12993 
.13007 
.13022 

•14896 
.14914 
•14933 
•14952 
•14971 

•13837 
•13852 
-13867 
•13881 
•13896 

-16059 
-16079 
.16099 
.16119 
.16139 

•14736 
.14751 
•14766 
•14782 
.14797 

•17283 
•17304 
-17325 
.17346 
.17367 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

•12188 
•12202 
•12216 
•12230 

.12244 

.13879 
.13897 
•13916 
.13934 
13952 

.13036 
.13051 
.13065 
.13079 
.  13094 

.14990 
.15009 
.15028 
•15047 
.15066 

•13911 
•13926 
•13941 
•13955 
.58970 

.16159 
•16179 
.16199 
•16219 
.16239 

.14812 

.14827 
.14843 
•14858 
•14873 

-17388 
•17409 
.  1  7430 
•17451 

•77472 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

•12257 
•12271 
•12285 
.12299 
.12313 

.13970 
.13988 
•14006 
.  14024 
.  14042 

-13108 
-13122 
.13137 
•13151 
.13166 

•15085 
•15105 
.15124 
•  15143 
.15162 

13985 
•  14000 
-14015 
•  14030 
•14044 

.16259 
.16279 
.16299 
•16319 
-16339 

.14888 
.  14904 
•14819 
.14934 
.14949 

•17493 
•17514 
•17535 
•17556 
.17577 

4O 

41 
42 
43 
44 

45 
48 
47 
48 
49 

•12327 
•]234i 
.12355 
.12369 
12383 

•14061 
•14079 
•14097 
•14115 
.14134 

•13180 
13195 
.13209 
.13223 
.13238 

.15181 
•15200 
•15219 
•15239 
.15258 

14059 
14074 
•14089 
•14104 
14119 

16359 
-16380 
-  16400 
-16420 
•16440 

•14965 
.14980 
-14995 
-15011 
-15026 

•17598 
•17620 
.17641 
.17662 
-176S3 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

•  1239*7 
•12411 
.12425 
.12439 
•12454 

.14152 
.14170 
•14188 
•14207 
.14225 

.13252 
.13267 
.13281 
•13296 
•13310 

.15277 
.15296 
.15315 
.15335 
•15354 

•14134 
•14149 
•14164 
•14179 
•14194 

.16460' 
16481 
.16501 
•16521 
.16541 

15041 
-15057 
-15072 
-15087 
-15103 

•17704 
•17726 
•17747 
•17768 
•17790 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

•12468 
•  12482 
•12496 
.12510 

.12524 

.14243 
14262 
•14280 
.14299 
•14317 

•13325 
.13339 
•13354 
.13368 
.13383 

•15373 
.15393 
.15412 
•15431 
.15451 

•14208 

•14223 
•14238 
-14253 
.14268 

.16562 
.16582 
.16602 
•16623 
-16643 

.15118 
•15134 
-15149 
-15164 
•15180 

•17811 
.17832 
.17854 
.17875 
•17896 

55 
56 
57 
58 

-59 

60 

.12538 

•14335 

.13397 

•15470 

-14283 

•16663 

•15195 

•17918 

60 

284    NATURAL  VERSED  SINES  AND  EXTERNAL  SEC. 

33°  33°  34°  35° 


• 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

' 

0 

1 

2 
3 
4 

.15195 
.15211 
•15226 
.15241 
-15257 

•17918 
•17939 
•17961 
•17982 
.  18004 

.16133 
•16149 
•16165 
•16181 
•16196 

•19236 
•19259 
•19281 
•19304 
•19327 

•17096 
•17113 
•17129 
.17145 
•17161 

•20622 
•20645 
•20669 
.20693 
•20717 

•18085 
-18101 
•18118 
•18135 
•18152 

-22077 
.22102 
.22127 
•22152 
•22177 

O 

1 

3 

4 

5 

7 
8 
0 

.15272 
.15288 
.15303 
.15319 
.15334 

.18025 
•18047 
•18068 
•18090 
•18111 

•16212 
.16228 
-16244 
•16260 
•16276 

.  19349 
•19372 
•19394 
•19417 
-19440 

•17178 
.17194 
•17210 
•17227 
•17243 

•20740 
.20764 
•20788 
•20812 
•20836 

•18168 
•18185 
•18202 
•18218 
•18235 

-  22202 
-22227 
-22252 
-22277 
-22302 

5 
6 
7 

8 

9 

iO 

11 
12 
13 
14 

.15350 
.15365 
.15381 
.15396 
.15412 

•18133 
.18155 
.18176 
.18198 
.18220 

•16292 
•16308 
•16324 
•16340 
-16355 

-19463 
-19485 
-19508 
.19531 
•19554 

•17259 
•17276 
•17292 
•17308 
•17325 

.20859 
•20883 
•20907 
•20931 
.20955 

•18252 
•18269 
•18286 
•18302 
•18319 

-22327 
-22352 
-22377 
.  22402 
•  22428 

IO 

11 
12 
13 
14 

15 
16 
17 
18 
19 

.15427 
.15443 
.15458 
•15474 
.15489 

.18241 
•18263 
•18285 
.18307 
.18328 

-16371 
•16387 
•16403 
•16419 

•16435 

•19576 
.19599 
.19622 
•19645 
•19668 

•17341 
•17357 
•17374 
-17390 

.17407 

•20979 
-21003 
-21027 
-21051 
-21075 

•18336 
•18353 
•18369 
•18386 

.18403 

.22453 
•22478 
.22503 
.22528 
•22554 

15 
16 
17 
18 

19 

20 

21 
22 
23 
24 

•15505 
•15520 
.15536 
•15552 
.15567 

.18350 
•18372 
.18394 
•18416 
.18437 

.16451 
•16467 
•16483 
•16499 
•16515 

.19691 
.19713 
-19736 
.19759 
•19782 

•17423 
•17439 
•17456 
•17472 
•17489 

-21099 
•21123 
-21147 
•21171 
.21195 

•18420 
•18437 
•18454 
•18470 
•18487 

-22579 
-22604 
-22629 
•22655 
•22680 

20 

21 

22 
23 
24 

25 
26 
27 
28 
29 

•15583 
.15598 
•15614 
•15630 
•15645 

.18459 
.18481 
.18503 
•18525 
.18547 

•16531 
•16547 
•16563 
•16579 
16595 

.19805 
.19828 
.19851 
•19874 
.19897 

•17505 
-17522 
•17538 
•17554 
-17571 

•21220 
•21244 
•21268 
•21292 
•21316 

•18504 
•18521 
•18538 
•18555 
.18572 

•22706 
•22731 
•22756 
•22782 
•22807 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

.15661 
.15676 
.15692 
•15708 
•15723 

.18569 
.18591 
.18613 
.18635 
.18657 

•16611 
•16627 
•16644 
•16660 
•16676 

.19920 
.19944 
•19967 
•19990 
.20013 

•17587 
•17604 
•17620 
•17637 
•17653 

•21341 
•21365 
•21389 
•21414 
.21438 

•18588 
•18605 
•18622 
•18639 
•18656 

•22833 
•22858 
•22884 
•22909 
.22935 

30 

31 
32 
33 
34 

35 
36 
87 
38 
39 

.15739 
.15755 
.15770 
.15786 
-15802 

.18679 
•18701 
•18723 
•18745 
•18767 

•16692 
•16708 
•16724 
•16740 

.16756 

.20036 
•20059 
-20083 
-20106 
.20129 

•17670 
•17686 
•17703 
•17719 
.17736 

•21462 
•21487 
.21511 
•21535 

•21560 

•18673 
•18690 
•18707 
•18724 
.18741 

.22960 
.22986 
•23012 
•23037 
•23063 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

.15818 
-15833 
.15849 
.15865 
•15880 

.18790 
•18812 
.18834 
•18856 
.18878 

•16772 
•16788 
•16805 
•16821 
.16837 

.20152 
•20176 
•20199 
•20222 
.20246 

•17752 
•17769 
•17786 
•17802 
.17819 

•21584 
•21609 
•21633 
•21658 
•21682 

-18758 
-18775 
•18792 
•18809 
•18826 

•23089 
•23114 
•23140 
•23166 
•23192 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

•15896 
.15912 
•15928 
•15943 
•15959 

.18901 
•18923 
•18945 
•18967 
.18990 

•16853 
•16869 
•16885 
•16902 
•16918 

.20269 
.20292 
.20316 
.20339 
.20363 

•17835 
•17852 
•17868 
•17885 
•17902 

•21707 
•21731 
-21756 
.21781 
.21805 

•18843 

.18860 
.18877 
•18894 
.18911 

•23217 
•23243 
•23269 
.23295 
.23321 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

•15975 
.15991 
•16008 
16022 
.18038 

.19012 
.19034 
.19057 
.19079 
.19102 

.16934 
•16950 
•16966 
•16983 
.16999 

•20386 
.20410 
.20433 
•20457 
.  20480 

•17918 
•17935 
•17952 
•17968 
•17985 

.21830 
.21855 
.21879 
•21904 
.21929 

•18928 
18945 
18962 
18979 
18996 

.23347 
.23373 
-23399 
•23424 
-23450 

50 

51 
52 
53 
54 

55 
58 
57 
58 
59 

16054 
16070 
16085 
16101 
16117 

.19124 
.19146 
.19169 
.19191 

•19214 

.17015 
.17031 
.17047 
•17064 
17080 

.20504 
•20527 
.20551 
.20575 
.20598 

•18001 
18018 
18035 
18051 
18068 

•21953 
.21978 
•22003 
.22028 
•22053 

19013 
19030 
19047 
19064 
19081 

•23476 
•23502 
•23529 
.23555 
.23581 

55 
56 
57 
58 
59 

60 

16133 

.19236 

.17096 

•20622 

18085 

-22077 

19098 

i 

•23607 

60 

NATURAL  VERSED  SINES  AND  EXTERNAL  SEC.    285 


36C 


37° 


38e 


39° 


* 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

t 

o 

1 

2 
3 
4 

.19098 
.19115 
•19133 
.19150 
.19167 

•23607 
•23633 
•23659 
•23685 
•23711 

•20136 
•20154 
.20171 
.20189 
•20207 

.25214 
•25241 
•25269 
•25296 
•  25324 

•21199 
•21217 
•21235 
.21253 
•21271 

•26902 
•26931 
.26960 
.26988 
•27017 

22285 
22304 
•22322 
22340 
•22359 

•28676 
•28706 
•28737 
•28767 
•28797 

-5 

2 

4 

5 
6 
7 

9 

.19184 
•19201 
•19218 
.19235 
•19252 

•23738 
.23764 
.23790 
•23816 
.23843 

.20224 
.20242 
.20259 
.20277 
.20294 

•25351 
•25379 
•25406 
•25434 
•25462 

•21289 
•21307 
•21324 
•21342 
.21360 

•  27046 
•27075 
•27104 
•27133 
•27162 

•22377 
•22395 
•22414 
•22432 
•22450 

•28828 
-28858 
.28889 
-28919 
-28950 

5 

8 

9 

10 

11 
12 
13 
14 

•19270 
19287 
19304 
19321 
.19338 

.23869 
•23895 
•23922 
•23948 
•23975 

•20312 
.20329 
•20347 
•20365 
.20382 

•25489 
.25517 
•25545 
•25572 
•25600 

•21378 
•21396 
•21414 
•21432 
•21450 

•27191 
•27221 
•27250 
•27279 
•27308 

•22469 
.22487 
•22506 
•22524 
•22542 

•28980 
.29011 
•  29042 
•29072 
•29103 

1O 

11 

12 
13 
14 

15 
16 
17 
18 

19 

19356 
19373 
19390 
19407 
19424 

.24001 
.24028 
•24054 
•24081 
.24107 

.  20400 
.20417 
.20435 
•20453 

.20470 

•25628 
.25656 
•25683 
•25711 
.25739 

•  21468. 
•21486 
•21504 
•21522 

.21540 

•27337 
•27366 
•27396 
•27425 

.27454 

.22561 
.22579 
•22598 
.22616 
•22634 

•29133 
•29164 
•29195 
•29226 

•29256 

15 
18 
17 
18 
19 

20 

21 
22 
23 
24 

•19442 
.19459 
.19476 
•  19493 
.19511 

.24134 
•24160 
•24187 
.24213 
•24240 

.20488 
•20506 
•20523 
.20541 
•20559 

.25767 
•25795 
•25823 
•25851 
•25879 

•21558 
•21576 
.21595 
•21613 
•21631 

.27483 
•27513 
•27542 
•27572 
•27601 

-22653 
•22671 
•22690 
•22708 
•22727 

•29287 
•29318 
•29349 
•29380 
.29411 

2O 

21 
22 
23 
24 

25 
26 
27 
28 
29 

.19528 
.19545 
•19562 
.19580 
.19597 

•24267 
.24293 
•24320 
•24347 
-24373 

•20576 
•20594 
.20612 
•20629 
.20647 

•25907 
•25935 
•25963 
•25991 
•26019 

•21649 
•21667 
•21685 
•21703 
•21721 

•27630 
-27660 
•27689 
•27719 
-27748 

•22745 
•22764 
•22782 
.22801 
•22819 

.  29442 
•29473 
•29504 
•29535 
•29566 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

•  19614 
•19632 
.19649 
•  19666 
•19684 

.  24400 
•24427 
.  24454 
•24481 
•24508 

.20665 
•20682 
•20700 
.20718 
•20736 

.26047 
•26075 
•26104 
•26132 
•26160 

•21739 
•21757 
•21775 
.21794 
•21812 

-27778 
-27807 
-27837 
-27867 
-27896 

•22838 

-22856 
22875 
•22893 
•22912 

•29597 
.29628 
•29659 
•29690 
•29721 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

.19701 
.19718 
•19736 
.19753 
.19770 

•24534 
•24561 
•24588 
-24615 
•24642 

.20753 
.20771 
.20789 
.20807 

.20824 

•26188 
•26216 
•26245 
•26273 
.26301 

•21830 
•21848 
•21866 
•21884 

.21902 

•27926 
.27956 
•27985 
•28015 

.28045 

•22930 
.22949 
•22967 
.22986 
-23004 

•29752 
•29784 
•29815 
•29846 

.29877 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

.19788 
.19805 
•19822 
•19840 
-19857 

•24669 
•24696 
•24723 
•24750 
.24777 

.20842 
•20860 
•20878 
.20895 
.20913 

.26330 
•26358 
•26387 
•26415 
•  26443 

.21921 
•21939 
•21957 
•21975 
•21993 

•28075 
•28105 
•28134 
•28164 
.28194 

•23023 
•23041 
.23060 
•23079 
.23097 

•29909 
•29940 
•29971 
-  30003 
•30034 

4O 

41 

42 
43 
44 

45 
46 
47 
48 
49 

.19875 
.19892 
.19909 
.19927 
•19944 

•  24804 
.24832 
24859 
•24886 
•24913 

•20931 
.20949 
•20967 
•20985 
.21002 

•26472 
.26500 
•26529 
.26557 
.26586 

•22012 
•22030 
•22048 
-22066 
.22084 

•28224 
•28254 
•28284 
•28314 
.28344 

•23116 
.23134 
•23153 
•23172 
.23190 

•30066 
•30097 
•30129 
•30160 
•30192 

45 
46 
47 
48 
49 

5O 

51 
52 
53 
54 

•19962 
.19979 
•19997 
•20014 
.20032 

.  24940 
•24967 
•24995 
•25022 
-25049 

•21020 
.21038 
•21056 
•21074 
.21092 

.26615 
•26643 
•26672 
.26701 
•26729 

•22103 
•22121 
.22139 
•22157 
•22176 

•28374  ' 
.  28404 
.28434 
.28464 
.28495 

•23209 
•23228 
.23246 
•23265 
•23283 

•30223 
•30255 
•30287 
•30318 
•30350 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

.  20049 
.20066 
•20084 
•20101 
•20119 

•25077 
.25104 
.25131 
•25159 
.25186 

•21109 
•21127 
•21145 
•21163 
.21181 

•26758 
•26787 
.26815 
•26844 
.26873 

•22194 
•22212 
•22231 
•22249 

.22267 

•28525 
.28555 
•28585 
•28615 
.28646 

•23302 
•23321 
.23339 
•23358 
-23377 

•30382 
•30413 
•30445 
•30477 
•30509 

55 
56 
57 
56 
59 

60 

.20136 

.25214 

•21199 

.26902 

•22285 

.28676 

•23396 

•30541 

60 

286  NATURAL  VERSED  SINES  AND  EXTERNAL  SEC. 

40°  41°  42°  43° 


/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

t 

o 

1 

2 
3 

4 

-23396 
•23414 
.23433 
-23452 
.23470 

•30541 
.30573 
.30605 
.30636 
.30668 

.24529 
.24548 
•24567 
.24586 
•24605 

.32501 
.32535 
.32568 
•32602 
.32636 

•25686 
.25705 
•25724 
•25744 
•25763 

•34563 
.34599 
.34634 
.34669 
.34704 

•26865 
•26884 
.26904 
•26924 
•26944 

•36733 
.36770 
.36807 
.36844 
.36831 

O 
1 

2 

a 

5 
6 
7 
8 

o 

.23489 
•23508 
•23527 
.23545 
.23564 

.30700 
•30732 
.30764 
.30796 
.30829 

.24625 
•  24644 
.24663 
.24682 
.24701 

.32669 
.32703 
.32737 
•32770 
.32804 

•25783 

.25802 
•25822 
•25841 
•25861 

.34740 
.34775 
.34811 
•34846 
•34882 

•26964 
•26984 
.27004 
•27024 
•27043 

.36919 
.36956 
.36993 
•37030 
.37068 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

.23583 
.23602 
•23620 
•23639 
.23658 

.30861 
.30893 
.30925 
.30957 
.30989 

.24720 
.24739 
.24759 
.24778 
.24797 

.32838 
.32872 
.32905 
.32939 
.32973 

.25880 

•25900 
•25920 
•25939 
•25959 

•34917 
.34953 
.34988 
•35024 
.35060 

27063 
•27083 
.27103 
.27123 
.27143 

.37105 
•37143 
•37180 
.37218 
•37255 

1O 

11 

12 
13 
14 

15 
16 
17 
18 

19 

.23677 
.23696 
•23714 
•23733 

-23752 

.31022 
.31054 
.31086 
.31119 
.31151 

.24816 
.24835 
.24854 
.24874 
.24893 

•33007 

.33041 
•33075 
.33109 
.33143 

.25978 
•25998 
•26017 
•26037 
•26056 

.35095 
.35131 
.35167 
.35203 
.35238 

.27163 
•27183 
•27203 
.27223 
•27243 

.37293 
.37330 
.37368 
.37406 
•37443 

15 
18 
17 
18 
19 

20 

21 
22 
23 
?A 

•23771 
.23790 
.23808 
.23827 
.23846 

.31183 
.31216 
.31248 
.31281 
.31313 

•24912 
.24931 
.24950 
.24970 
.24989 

.33177 
.33211 
.33245 
.33279 
.33314 

.26076 
•26096 
.26115 
.26135 
.26154 

•35274 
.35310 
•35346 
.35382 
.35418 

.27263 
.27283 
.27303 
.27323 
.27343 

•37481 
.37519 
.37556 
.37594 
.37632 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

.23865 
-23884 
.23903 
-23922 
-23941 

.31346 
.31378 
.31411 
.31443 
.31476 

.25008 
.25027 
.25047 
.25066 
.25085 

.33348 
.33382 
.33416 
.33451 
.33485 

.26174 
.26194 
.26213 
.26233 
.26253 

.35454 
.35490 
.35526 
•35562 
.35598 

•27363 
•27383 
•27403 
•  27423 
.  27443 

.37670 
.37708 
.37746 
.37784 
•37822 

25 
26 
27 
28 

29 

30 

31 
32 
33 
34 

.23959 
.23978 
.23997 
.24016 
-24035 

•31509 
•31541 
.31574 
.31607 
.31640 

.25104 
.25124 
.25143 
.25162 
.25182 

.33519 
.33554 
.33588 
.33622 
.33657 

.26272 
.26292 
.26312 
•26331 
.26351 

.35634 
•35670 
.35707 
.35743 
.35779 

.27463 
.27483 
.27503 
.27523 
.27543 

.37860 
.37898 
.37936 
.37974 
.38012 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

.24054 
-24073 
-24092 
.24111 
.24130 

.31672 
.31705 
.31738 
.31771 
.31804 

.25201 
.25220 
.25240 
.25259 
.25278 

.33691 
.33726 
.33760 
.33795 
-33830 

.26371 
•26390 
•26410 
•26430 
•26449 

.35815 

.35852 
.35888 
.35924 
•35961 

.27563 
.27583 
.27603 
.27623 
.27643 

.38051 
.38089 
.38127 
.38165 
.38204 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

-24149 
.24168 
.24187 
.24206 
-24225 

.31837 
.31870 
.31903 
.31936 
.31969 

.25297 
.25317 
.25336 
.25356 
.25375 

.33864 
.33899 
.33934 
.33968 
.34003 

•26469 
•26489 
.26509 
.26528 
.26548 

.35997 
.36034 
.36070 
.36107 
.36143 

•27663 
.27683 
•27703 
•27723 
.27743 

.38242 
.38280 
.38319 
.38357 
•38396 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

.  24244 
.24262 
.24281 
•24300 
-24320 

.32002 
.32035 
.32068 
.32101 
.32134 

.25394 
•25414 
.25433 
.25452 
.25472 

.34038 
.34073 
.34108 
.34142 
.34177 

.26568 
•26588 

.26607 
.26627 
.26647 

.36180 
.36217 
.36253 
.36290 
•36327 

•27764 
.27784 
.27804 
•27824 
.27844 

•38434 
.38473 
.38512 
.38550 
•38589 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

•24339 
.24358 
.24377 
.24396 
•244T5 

.32168 
.32201 
.32234 
.32267 
.32301 

.25491 
.25511 
•25530 
•25549 
.25569 

•34212 
.  34247 
.34282 
.34317 
.34352 

.26667 
-26686 
•26706 
.26726 
.26746 

•36363 
.36400 
.36437 
.36474 
•36511 

•27864 
.27884 
.27905 
.27925 
.27945 

.38628 
•38666 
.38705 
.38744 
.38783 

50 

51 
52 
53 
54 

55 
56 
57 
58 

59 

.  24434 
.24453 
.24472 
.24491 
-24510 

.32334 
.32368 
.32401 
.32434 
•32468 

.25588 
.25608 
.25627 
.25647 
.25666 

.34387 

.34423 
.34458 
.34493 
.34528 

•26766 
.26785 
.26805 
.26825 
.26845 

.36548 
•36585 
.36622 
•36669 
•36696 

.27965 
.27985 
28005 
.28026 
.28046 

.38822 
.38860 
•38899 
•38938 
•38977 

55 
56 
57 
58 
59 

60 

.24529 

.32501 

.25686 

.34563 

.26865 

.36733 

.28066 

.39016 

6O 

NATURAL  VERSED   SINES  AND  EXTERNAL  SEC.    287 
44°  45°  46°  47° 


1 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

' 

0 

1 

2 
3 
4 

•28066 
.28086 
.28106 
•28127 
•28147 

.39016 
.39055 
.39095 
.39134 
.39173 

-29289 
.29310 
•29330 
•29351 
.29372 

.41421 
.41463 
.41504 
.41545 
•41586 

•30534 
•30555 
.30576 
•30597 
•30618 

.43956 
.43999 
.44042 
.44086 
•44129 

•31800 
.31821 
•31843 
•31864 
.31885 

•46628 
•46674 
•46719 
•46765 
•46811 

0 

1 
2 
3 

4 

5 
6 
7 
8 
9 

.28167 
.28187 
.28208 
.28228 
•  28248 

.39212 
.39251 
.39291 
.39330 
•39369 

•29392 
.29413 
•29433 
.29454 
•29475 

.41627 
.41669 
.41710 
•41752 
•41793 

.30639 
.30660 
•30681 
.30702 
.30723 

•44173 
.44217 
.44260 
.44304 
.44347 

.31907 
•31928 
•31949 
•31971 
.31392 

•46857 
•46903 
•46949 
.46995 
-47041 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

.28268 
•28289 
.28309 
.28329 
.28350 

.39409 
.39448 
•39487 
.39527 
•39566 

.29495 
.29516 
•29537 
•29557 
•29578 

.41835 
.41876 
.41918 
•41959 
•42001 

.30744 
.30765 
.30786 
.30807 
.30828 

.44391 
•44435 
.44479 
.44523 
•44567 

•32013 
.32035 
•32056 
-32077 
-32099 

•47087 
.47134 
•47180 
.47226 
-47272 

10 

11 
12 
13 
14 

15 
16 
17 
18 

19 

•28370 
.28390 
.28410 
•28431 

•  28451 

.39606 
.39646 
.39685 
.39725 
•39764 

.29599 
.29619 
•29640 
•29661 
•29681 

.  42042 
.42084 
.42126 
•42168 
-42210 

.30849 
.30870 
-30891 
-30912 
-30933 

•44610 
.44654 
.44698 
.44742 
-44787 

•32120 
.32141 
•32163 
-32184 
•32205 

•47319 
.47365 
•47411 
.47458 

.47504 

15 
16 
17 
18 
.19 

20 

21 
22 
23 
24 

.28471 
.28492 
.28512 
.28532 
.28553 

•39804 
.39844 
.39884 
.39924 
-39963 

•29702 
.29723 
•29743 
.29764 
•29785 

•42251 
.42293 
.42335 
.42377 
.42419 

-30954 
-30975 
-30996 
-31017 
.31038 

•44831 
.44875 
.44919 
.44963 
•45C07 

-32227 
•32248 
•32270 
•32291 
•32312 

•47551 
•47598 
.47644 
•47691 
-47738 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

.28573 
.28593 
.28614 
.28634 
•  28655 

.  40003 
.40043 
.40083 
.40123 
.40163 

.29805 
.29826 
.29847 
•29868 
-29888 

.42461 
.42503 
.42545 
.42587 
.42630 

.31059 
•31080 
•31101 
•31122 
-31143 

.45052 
.45096 
.45141 
.45185 
•45229 

•32334 
•32355 
•32377 
•32398 
•32420 

.47784 
•47831 
•47878 
•47925 
-47972 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

.28675 
.28695 
.28716 
.28736 
-28757 

.40203 
.40243 
.40283 
.40324 
•40364 

.29909 
•29930 
.29951 
.29971 
-29992 

.42672 
.42714 
.42756 
.42799 
.42841 

•31165 
.31186 
•31207 
•31228 
.31249 

.45274 
.45319 
•45363 
.45408 
.45452 

•32441 
-32462 
•32484 
•32505 
•32527 

-48019 
•48066 
.48113 
•48160 
-48207 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

.28777 
•28797 
•  28818 
.28838 
.28859 

.  40404 
.40444 
.40485 
.40525 
.40565 

•30013 
•30034 
-30054 
•30075 
-30096 

•42883 
•42926 
.42968 
•43011 

-43053 

•31270 
-31291 
-31312 
.31334 
.31355 

•45497 
.45542 
.45587 
.45631 
-45676 

.32548 
-32570 
-32591 
•32613 
.32634 

•48254 
•48301 
.48349 
•48396 

.48443 

35 
36 
37 
38 

39 

40 

41 
42 
48 
44 

.28879 
.28900 
.28920 
•28941 
.28961 

.40606 
.40646 
.40687 
.40727 
.40768 

•30117 
•30138 
.30158 
.30179 
•30200 

.43096 
.43139 
.43181 
•43224 
•43267 

•31376 
•31397 
•31418 
.31439 
.31461 

•45721 
.45766 
.45811 
.45856 
•45901 

•32656 
-32677 
-32699 
•32720 
-32742 

•48491 
•48538 
•48586 
•48633 
•48681 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

.28981 
•29002 
•29022 
.29043 
.29063 

.40808 
.40849 
.40890 
.40930 
.40971 

•30221 
-30242 
-30263 
.30283 
.30304 

.43310 
.43352 
.43395 
•43438 
.43481 

-31482 
•31503 
•31524 
•31545 
.31567 

.45946 
•45992 
-46037 
46082 
-46127 

-32763 
-32785 
•32806 
-32828 
.32849 

.48728 
.48776 
•48824 
•48871 
48919 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

•  29084 
•29104 
.29125 
.29145 
•29166 

.41012 
.41053 
.41093 
.41134 
.41175 

-30325 
-30346 
-30367 
-30388 
.  30409 

.43524 
.43567 
.43610 
.43653 
.43696 

•31588 
•31609 
•31630 
•31651 
•31673 

.46173 
•46218 
•46263 
•46309 
•46354 

•32871 
•32893 
•32914 
•32936 
-32P57 

•48967 
•49015 
.49063 
.49111, 
.49159 

50 

51 
52 
53 
54 

55 
56 
57 
58 
5fl 

•29187 
•29207 
•29228 
•  29248 

•29269 

.41216 
.41257 
.41298 
.41339 
.41380 

•30430 
.30451 
•30471 
-30492 
-30513 

.43739 
.43783 
.43826 
-43869 
.43912 

•31694 
•31715 
.31736 
.31758 
.31779 

.46400 
.46445 
.46491 
.46537 
.46582 

•32979 
33001 
33022 
33044 
3SOB5 

.49207 
•49255 
.49303 
•49351 
.49399 

55 
56 
57 
58 
59 

60 

.29289 

.41421 

.30534 

•43956 

•31800 

.46628 

33087 

•49448 

60 

288    NATURAL  VERSED  SINES  AND  EXTERNAL  SEC. 

48°  49°  50°  51° 


• 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

i 

0 

2 
4 

33087 
33109 
33130 
33152 
33173 

.49448 
.49496 
.49544 
.49593 
.49641 

34394 
34416 
34438 
34460 
34482 

.52425 
.52476 
.52527 
.52579 
.52630 

35721 
35744 
35766 
35788 
35810 

•55572 
•55626 
•55680 
.55734 
•55789 

37068 
37091 
37113 
37136 
37158 

•58902 
.58959 
•59016 
•59073 
•59130 

O 

1 
2 
3 
4 

5 
6 
7 
8 
9 

33195 
33217 
33238 
33260 
33282 

.49690 
•49738 
.49787 
•49835 
.49884 

34504 
34526 
34548 
34570 
34592 

•52681 
.52732 
.52784 
.52835 
.52886 

35833 
35855 
35877 
35900 
35922 

.55843 
.55897 
.55951 
•56005 
•56060 

37181 
37204 
37226 
37249 
37272 

•59188 
•59245 
•59302 
•59360 
-59418 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

33303 
33325 
33347 
33368 
33390 

.49933 
•49981 
.50030 
.50079 
.50128 

34614 
34638 
34658 
34680 
34702 

•52938 
.52989 
.53041 
.53092 
.53144 

35944 
35967 
35989 
36011 
36034 

•56114 
.56169 
.56223 
.56278 
.56332 

37294 
37317 
37340 
37362 
37385 

•59475 
.59533 
•59590 
•59648 
.59706 

10 

11 

12 
13 
14 

15 
16 
17 
18 
19 

33412 
33434 
33455 
33477 
33499 

.50177 
.50226 
•50275 
•50324 
•50373 

34724 
34748 
34768 
34790 

34812 

•53196 
.53247 
.53299 
•53351 

.53403 

36056 
36078 
36101 
36123 
3S146 

•56387 

.  56442 
•56497 
.56551 
.56606 

37408 
37430 
37453 
37476 
37498 

.59764 
•59822 
.59880 
•59938 
.59996 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

33520 
33542 
33584 
33586 
33807 

•50422 
•50471 
.50521 
.50570 
.50819 

34834 
34856 
34878 
34900 
34923 

.53455 
•53507 
•53559 
.53611 
.53663 

3G168 
36190 
36213 
36235 
38258 

•56661 
•56716 
.56771 
.56826 
.56881 

37521 
37544 
37567 
37589 
37612 

.60054 
•60112 
•60171 
.60229 
•60287 

30 

21 

22 
23 
24 

25 
26 
27 
28 
29 

33629 
33651 
33873 
33694 
33716 

•50669 
.50718 
.50767 
.50817 
.50866 

34945 
34967 
34989 
35011 
35033 

•53715 
.53768 
.53820 
.53872 
.53924 

36280 
36302 
36325 
36347 
36370 

.56937 
.56992 
.57047 
.57103 
.57158 

37635 
37658 
37680 
37703 
37726 

.60346 
.60404 
•60463 
•60521 
.60580 

25 
28 
27 
28 
29 

30 

31 
32 
33 
34 

33738 
33760 
33782 
33803 
33825 

.50916 
•50966 
•51015 
.51085 
.51115 

35055 
35077 
35099 
35122 
35144 

.53977 
•54029 
.54082 
•54134 
.54187 

36392 
36415 
36437 
36460 
36482 

.57213 
•57269 
•57324 
•57380 
.57436 

37749 
37771 
37794 
37817 
.37840 

•60639 
•60698 
•60756 
•60815 
•60874 

30 

31 
32 
33 
34 

35 
38 
37 
38 
39 

33847 
33869 
33891 
33912 
.33934 

.51165 
.51215 
.51265 
.51314 
.51364 

35166 
35188 
35210 
35232 
35254 

.  54240 
•54292 
•  54345 
.54398 

.54451 

36504 
36527 
36549 
36572 
36594 

•57491 
.57547 
.57603 
.57659 
.57715 

.37862 
•37885 
•37908 
.37931 
.37954 

.60933 
.60992 
.61051 
.61111 
.61170 

35 
36 
37 
38 
-39 

40 

41 
42 
43 
44 

•33956 
.33978 
.34000 
.34022 
.34044 

.51415 
.51465 
•51515 
.51585 
.51615 

35277 
35299 
35321 
35343 
35365 

.54504 
.54557 
•54610 
.54663 
.54716 

36617 
36639 
•36662 
•36684 
•36707 

•57771 
.57827 
.57883 
.57939 
•57995 

.37976 
•37999 
•38022 
.38045 
.38068 

.61229 
.61288 
•61348 
.61407 
.61467 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

.34065 
-34087 
.34109 
•34131 
.34153 

.51665 
•51716 
.51768 
.51817 
.51867 

35388 
35410 
35432 
-35454 
.35478 

•54769 
.54822 
•54876 
-  54929 
•54982 

•36729 
•36752 
•36775 
•36797 
.36820 

.58051 
.58108 
.58164 
.58221 
.58277 

•38091 
•38113 
•38136 
•38159 
.38182 

.61526 
.61586 
•61646 
.61705 
.61765 

45 
48 
47 
48 
49 

50 

51 
52 
53 
54 

•34175 
•34197 
•34219 
•34241 
•34282 

•51918 
.51968 
.52019 
.52069 
•52120 

•35499 
-35521 
•35543 
•35565 
•35588 

•55036 
.55089 
•55143 
.55198 
.55250 

•36842 
•36865 
•36887 
•36910 
•36932 

.58333 
.58390 
.58447 
.58503 
.58560 

.38205 
•38228 
•38251 
•38274 
•38298 

.61825 
.61885 
•61945 
•62005 
•62065 

50 

51 

n 

54 

55 
56 
57 
58 
59_ 

•34284 
•34308 
•34328 
•34350 
-34372 

.52171 
.52222 
.52273 
.52323 
.52374 

.35610 
•35632 
.35654 
.35677 
•35699 

.55303 
.55357 
.55411 
.55465 
.55518 

.36955 
•36978 
•37000 
•37023 
•37045 

.58617 
.58674 
•58731 
•58788 
•58845 

•38319 
•38342 
.38365 
.38388 
•38411 

•62125 
.62185 
.62246 
•62306 

.62366 

55 
56 
57 
58 
59 

60 

•34394 

.52425 

•35721 

.55572 

•37068 

•58902 

•38434 

.62427 

CO 

NATURAL  VERSED   SINES  AND  EXTERNAL  SEC.    289 

53°  53°  54°  55° 


/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

) 

0 

1 

2 
3 
4 

•38434 
.38457 
.38480 

.38503 
.38526 

.62427 
.62487 
.  62548 
.62609 
.  62669 

.39819 
.  39842 
.39865 
•39888 
•39911 

.66164 
.66228 
.66292 
•66357 
•66421 

.41221 
.41245 
.41269 
-41292 
.41316 

.70130 
.70198 

.70267 
•70335 
.  70403 

•42642 
•42666 
-42690 
-42714 
•42738 

.  74345 
•74417 
•  74490 
•74562 
•74635 

0 

1 
2 
3 

4 

5 
6 
7 
8 
9 

•38549 
•38571 
•38594 
•38617 
•38640 

.62730 
.62791 
.62852 
.62913 
•62974 

39935 
39958 
39981 
40005 
40028 

.66486 
.66550 
.66615 
•66679 
•  66744 

.41339 
-41363 
.41386 
-41410 
-41433 

.70472 
•70540 
.70609 
•70677 
.  70746 

-42762 
-42785 
•42809 
-42833 
-42857 

•  74708 
.74781 
•  74854 
•  74927 
.75000 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

•38663 
•38686 
•38709 
•38732 
•38755 

•63035 
•63096 
.63157 
.63218 
.63279 

40051 
40074 
40098 
40121 
40144 

.66809 
•66873 
-66938 
•67003 
.67068 

.41457 
•41481 
-41504 
-41528 
-41551 

70815 
•70884 
•70953 
•71022 
.71091 

•42881 
-42905 
•42929 
•42953 
•42976 

•75073 
•75146 
.75219 
.75293 
-75366 

10 

11 
12 
13 

14 

15 
16 
17 
18 

IL. 

.38778 
•38801 
•38824 
.38847 
.38870 

•63341 
•63402 
.63464 
•63525 
•63587 

40168 
40191 
.40214 
.40237 
.40261 

.67133 
.67199 
.67264 
.67329 
.67394 

.41575 
.41599 
.41622 
.41646 
.41670 

•71160 
•71229 
•71298 
.71368 
•71437 

•43000 
-43024 
•43048 
-43072 
.43096 

.  75440 
.75513 
.75587 
.75661 
•75734 

15 
16 
17 
18 
J9 

30 

21 
22 
23 
,24 

.38893 
.38916 
•38939 
.38962 
.38985 

•63648 
.63710 
.63772 
•63834 
•63895 

.40284 
.40307 
.40331 
.40354 
.40378 

•67460 
.67525 
•67591 
•67656 
.67722 

•41693 
.41717 
-41740 
.41764 
-41788 

•71506 
.71576 
•71646 
.71715 
.71785 

•43120 
.43144 
.43168 
-43192 
43216 

.75808 
•75882 
.75956 
.76031 
•76105 

2O 

21 
22 
23 
24 

25 
26 
27 
28 
29 

.39009 
.39032 
.39055 
.39078 
•39101 

•63957 
.64019 
•64081 
•64144 
.64208 

.40401 
.40424 
•  40448 
•40471 
.40494 

.67788 
.67853 
•67919 
.67985 
•68051 

.41811 
.41835 
.41859 
.41882 
.41906 

.71855 
.71925 
•71995 
.72065 
•72135 

•43240 
•43264 
-43287 
-43311 
-43335 

•76179 
•76253 
•76328 
.  76402 
•76477 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

.39124 
•39147 
.39170 
.39193 
.39216 

.64268 
.64330 
.64393 
.64455 
•64518 

-40518 
•40541 
•40565 
•40588 
.40611 

.68117 
.68183 
.68250 
.68316 
.68382 

.41930 
.41953 
.41977 
•42001 
-42024 

.  72205 
.  72275 
.72346 
.72416 
•  72487 

•43359 
.43383 
.  43407 
•43431 
-43455 

•76552 
.76626 
.76701 
.76776 
•76851 

30 

31 
32 
33 
34 

35 
36 
37 
38 
SSL 

.39239 
.39262 
•39286 
•39309 
.39332 

.64580 
•64643 
.64705 
.64768 
.64831 

.40635 
.40658 
•40682 
.40705 
-40728 

•68449 
•68515 
.68582 
.68648 
-68715 

-42048 
-42072 
-42096 
-42119 
-42143 

•72557 
.72628 
.  72698 
.72769 
.  72840 

•43479 
-43503 
-43527 
•43551 
.43575 

•76926 
.77001 
.77077 
.77152 
•77227 

35 
36 
37 
38 
39 

40 

41 
42 
43 
,44 

.39355 
•39378 
.39401 
.39424 
.39447 

.  64894 
•64957 
.65020 
.65083 
.65146 

.40752 
.40775 
.40799 
•40822 
-40846 

•68782 
.68848 
•68915 
.68982 
-69049 

-42167 
.42191 
.42214 
.42238 
-42262 

.72911 
.72982 
.73053 
.73124 
.73195 

-43599 

-43623 
•43647 
-43671 
.43695 

.77303 
.77378 
•77454 
•77530 
•77606 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

•39471 
.39494 
.39517 
-39540 
•39563 

•65209 
.65272 
.65336 
.65399 
.65462 

.40869 
.40893 
.40916 
-40939 
.40963 

.69116 
.69183 
.69250 
.69318 
.69385 

.42285 
.42309 
.42333 
.42357 
.42381 

.73267 
.73338 
.73409 
.73481 
.73552 

•43720 
•43744 
•43768 
•43792 
•43816 

.77681 
.77757 
.77833 
.77910 
•77986 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

-39586 
•39610 
.39633 
.39656 
.39679 

.65526 
.65589 
.65653 
.65717 
.65780 

-40986 
.41010 
•41033 
.41057 
.41080 

.  69452 
.69520 
.69587 
.69655 
.69723 

.42404 
.42428 
.42452 
.42476 
.42499 

.73624 
.73696 
.73768 
.73840 
•73911 

•43840 
.43864 
.43888 

.43912  ' 
-43936 

•78062 
.78138 
.78215 
.78291 
.78368 

50 

51 
52 
53 
54 

55 
56 
57 
58 
5J 

.39702 
•39726 
.39749 
.39772 
-39795 

.  65844 
•65C08 
.65972 
.66036 
.66100 

.41104 
.41127 
.41151 
•41174 
.41198 

.89790 
.69858 
.69926 
.69994 
.70062 

•42523 
.42547 
.42571 
.42595 
.42619 

.73983 
.74056 
.74128 
.  74200 
.74272 

-43960 
-43984 
-44008 
.44032 
.44057 

.  78445 
.78521 
.78598 
.78675 
.78752 

55 
56 
57 
58 
59 

60 

.39819 

.66164 

.41221 

.70130 

.42642 

.74345 

.44081 

.78829 

6Q 

290    NATURAL  VERSED  SINES  AND  EXTERNAL  SEC. 
56°  57°  58°  59° 


f 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

1 

0 

1 

2 
3 
4 

.44081 
.44105 
.44129 
•44153 
.44177 

.78829 
.78906 
-78984 
.79061 
•79138 

-45536 
-45560 
-45585 
•45609 
-45634 

.83608 
•83690 
.83773 
.83855 
.83938 

-47008 
-47033 
-47057 
-47082 
•47107 

•88708 
•88796 
•88884 
•88972 
-89060 

-48496 
-48521 
-48546 
.48571 
-48596 

•94160 
•94254 
.94349 
-  94443 
•94537 

0 

1 

2 
3 
4 

5 
6 
7 
8 
9 

.44201 
.44225 
.44250 
•44274 
44298 

•79216 
•79293 
.79371 
.79449 
.79527 

-45658 
-45683 
-45707 
-45731 
•45756 

•84020 
•84103 
•84186 
.84269 
.84352 

-47131 
•47156 
•47181 
-47206 
-47230 

•89148 
•89237 
.89325 
•89414 
-89503 

-48621 
-48646 
-48671 
-48696 
-48721 

•94632 
•94726 
•94821 
•94916 
•95011 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

.44322 
.44346 
.44370 
-44395 
-44419 

•79604 
•79682 
.79761 
.79839 
.79917 

-45780 
-45805 
45829 
45854 
.45878 

•84435 
•84518 
•84601 
•84685 
.84768 

-47255 
-47280 
•47304 
•47329 
•47354 

•89591 
•89680 
•89769 
•89858 
•89948 

-48746 
-48771 
-48796 
-48821 
-48846 

-95106 
-95201 
-95296 
•95392 
.95487 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

-  44443 
-44467 
•44491 
-44516 
-44540 

.79995 
.80074 
.80152 
.80231 
-80309 

•45903 
-45927 
-45951 
-45976 
-46000 

•84852 
.84935 
.85019 
•85103 
-85187 

•47379 
•47403 
•47428 
•47453 
-47478 

-90037 
-90126 
-90216 
.90305 
-90395 

•48871 
•48896 
•48921 
•48946 
-48971 

•95583 
•95678 
•95774 
•95870 

•95966 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

-44564 
-44588 
-44612 
-44637 
-44661 

.80388 
•80467 
.80546 
•80625 
.80704 

-46025 
-46049 
-46074 
-46098 
.46123 

•85271 
•85355 
•85439 
•85523 
-85608 

-47502 
•47527 
-47552 
-47577 
-47601 

-90485 
•90575 
.90665 
.90755 
•90845 

.48996 
•49021 
-49046 
•49071 
.49096 

•96062 
•96158 
•96255 
.96351 
•96448 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

-44885 
-44709 
-44734 
-44758 
-44782 

.80783 
•80862 
.80942 
.81021 
•81101 

•46147 
•46172 
-46196 
-46221 
-46246 

.85692 
•85777 
.85861 
•85946 
.86031 

-47626 
-47651 
-47676 
-47701 
-47725 

•90935 
•91026 
•91116 
.  91207 
•91297 

-49121 
.49146 
-49171 
-49196 
-49221 

.96544 
.96641 
.96738 
.96835 
.96932 

25 
26 
27 
28 

20 

30 

31 
32 
33 
34 

-44806 
•44831 
-44855 
-44879 
-44903 

.81180 
•81260 
•81340 
•81419 
.81499 

-46270 
-46295 
-46319 
•46344 
•46368 

.86116 
•86201 
•86286 
.86371 
•86457 

-47750 
-47775 
-47800 
.47825 
•47849 

•91388 
.91479 
•91570 
.91661 
•91752 

-49246 
-49271 
-49296 
49321 
.49346 

•97029 
.97127 
.97224 
.97322 
•97420 

30 

3] 

32 
33 
34 

35 
36 
37 
38 
39 

•44928 
-44952 
-44976 
-45001 

-45025 

•81579 
•81659 
•81740 
.81820 

81900 

•46393 
•46417 
•46442 
.46466 
•46491 

•86542 
.86627 
•86713 
•86799 
-86885 

•47874 
.47899 
-47924 
-47949 
-47974 

•91844 
.91935 
•92027 
.92118 

-92210 

•49372 
.49397 
•49422 
-49447 
-49472 

•97517 
.97615 
.97713 
.97811 

.97910 

35 
36 
37 
38 

39 

40 

41 
42 
43 
44 

-45049 
-45073 
-45098 
•45122 
-45146 

-81981 
•82061 
.82142 
•82222 
.82303 

-46516 
-46540 
-46565 
.46589 
.46614 

•86970 
•87056 
.87142 
•87229 
-87315 

-47998 
-48023 
-48048 
-48073 
-48098 

.92302 
•92394 
•92486 
•92578 
.92670 

-49497 
-49522 
-49547 
-49572 
.49597 

.98008 
•98107 
.98205 
.98304 
.98403 

40 

41 

42 
43 
44 

45 
46 
47 
48 
49 

-45171 
-45195 
-45219 
-45244 
.45268 

.82384 
.82465 
-82546 
•82627 
.82709 

•46639 
-46663 
-46688 
-46712 
-46737 

•87401 
.87488 
-87574 
-87661 
.87748 

•48123 
•48148 
-48172 
-48197 
-48222 

•92762 
•92855 
.92947 
•93040 
•93133 

.49623 
-49648 
-49673 
-49698 
-49723 

•98502 
.98601 
.98700 
.98799 
.98899 

45 
46 
47 
48 
49 

SO 

51 
52 
53 
54 

-45292 
-45317 
-45341 
-45365 
-45390 

•82790 
•82871 
.82953 
•83034 
.83116 

-46762 
-46786 
-46811 
-46836 
.46860 

.87834 
•87921 
•88008 
•88095 
-88183 

-48247 
-48272 
-48297 
-48322 
-48347 

.93226 
•93319 
.93412 
•93505 
.93598 

-49748 
-49773 
-49799 
-49824 
-49849 

•98998 
•99098 
•99198 
•99298 
-99398 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

-45414 
-45439 
-45463 
-45487 
-45512 

.83198 
.83280 
.83362 
•83444 
-83526 

.46885 
•46909 
-46934 
.46959 
-46983 

•88270 
.88357 
•88445 
.88532 
.88620 

.48372 
•48396 
•48421 
.48446 
-48471 

.93692 
•93785 
•93879 
•93973 
-94066 

•49874 
.49899 
•49924 
•49950 
-49975 

.99498 
•99598 
•99698 
.99799 
-99899 

55 
56 
57 
58 
59 

60 

-45536 

.83608 

•47008 

.88708 

•48496 

.94160 

•50000 

1-00000 

60 

NATURAL  VERSED  SINES  AND  EXTERNAL  SEC. 
60°  61fc  63°  63° 


291 


' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec 

' 

0 

1 

2 
3 
4 

•5UOOO 
•50025 
.50050 
-50076 
•50101 

i.  00000 
1-00101 
1-00202 
1-00303 
1  •  00404 

•51519 
-51544 
-51570 
-51595 
.51621 

1-06267 
1-06375 
1-06483 
1-06592 
1-06701 

-53053 
-53079 
•53104 
.53130 
.53156 

1.13005 
1-13122 
1-13239 
1.13356 
1-13473 

-54601 
-54627 
-54653 
•54679 
•547C5 

1-20269 
1.20395 
1.20521 
1-20647 
1-20773 

0 
1 

2 
3 
4 

5 
6 
7 
8 
9 

-50126 
-50151 
.50176 
-50202 
-50227 

1-00505 
1-00607 
1-00708 
1-00810 
1.00912 

-51646 
-51672 
-51697 
-51723 
-51748 

1-06809 
1-06918 
1.07027 
1.07137 
1-07246 

•53181 
-53207 
-53233 
•53258 
-53284 

1-13590 
1.13707 
1-13825 
1-13942 
1.14060 

•  54731 
.54757 
•  54782 
•  54808 
-54834 

1-20900 
1-21026 
1-21153 
1-21280 
1-21407 

5 
6 
7 
8 
9 

10 

11 

12 
13 
14 

•50252 
.50277 
-50303 
•50328 
.50353 

1-01014 
1-01116 
1-01218 
1-01320 
1.01422 

-51774 
-51799 
-51825 
-51850 
.51876 

1-07356 
1-07465 
1-07575 
1.07685 
1-07795 

-53310 
•53336 
-53361 
-53387 
-53413 

1-14178 
1-14296 
1-14414 
1-14533 
1.14651 

.54860 
•54886 
-54912 
-54938 
-54964 

1-21535 
1-21662 
1-21790 
1-21918 
1.22045 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

.50378 
.  50404 
•50429 
•50454 
-50479 

1-01525 
1-01628 
1-01730 
1-01833 
1.01936 

•51901 
•51927 
.51952 
•51978 

.52003 

1-07905 
1-08015 
1-08126 
1-08236 
1-08347 

•  53439 
.  53464 
-53490 
-53516 

.53542 

1-14770 
1.14889 
1-150C8 
1-15127 
1-15246 

.  54990 
-55016 
-  55042 
.55068 
.55094 

1-22174 
1-22302 
1-22430 
1-22559 
1.22688 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

-50505 
-50530 
-50555 
.50581 
.50606 

1-02039 
1-02143 
1.02246 
1-02349 
1-02453 

•52029 
•52054 
-52080 
•52105 
.52131 

1-08458 
1-08569 
1.08680 
1-08791 
1.08903 

-53567 
-53593 
-53619 
•53645 
•53670 

1-15366 
1-15485 
1-15605 
1-15725 
1-15845 

•55120 
55146 
-55172 
-55198 
-55224 

1-22817 
1-22946 
1-23075 
1.23205 
1-23334 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

-50631 
-50656 
-50682 
.50707 
.50732 

1-02557 
1.02661 
1-02765 
1-02869 
1-02973 

-52156 
-52182 
-52207 
-52233 
-52259 

1-09014 
1-09126 
1-09238 
1-09350 
1-09462 

•53696 
•53722 
•53748 
•53774 
•53799 

1-15965 
1-16085 
1-16206 
1-16326 
1-16447 

-55250 
-55276 
-55302 
.55328 
-55354 

1  -  23464 
1-23594 
1  •  23724 
1.23855 
1-23985 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

•50758 
-50783 
-50808 
-50834 
.50859 

1.03077 
1-03182 
1-03286 
1-03391 
1-03496 

.52284 
.52310 
•52335 
•52361 
•52386 

1-09574 
1-09686 
1-09799 
1-09911 
1-10024 

.53825 
•53851 
•53877 
•53903 
.53928 

1-16568 
1-16689 
1-16810 
1-16932 
1-17053 

-55380 
-55406 
-55432 
•55458 
•55484 

1-24116 
1-24247 
1-24378 
1-24509 
1-24640 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

.50884 
.50910 
.50935 
.50960 
50980 

1-03601 
1-03706 
1-03811 
1-03916 
1-04022 

52412 
52438 
52463 
52489 

52514 

1.10137 
1-10250 
1-10363 
1-10477 
1-10590 

•53954 
•53980 
•  54006 
•54032 
.  54058 

1.17175 
1-17297 
1-17419 
1-17541 
1  -17663 

•55510 
•55536 
•55563 
-E5589 

•EE615 

1-24772 
1-24903 
1-25035 
1-25167 
1  .  ?53CO 

35 
36 
37 
38 
.  39 

40 

41 
42 
43 
44 

-51011 
51036 
51062 
-51087 
51113 

1-04128 
1-04233 
1-04339 
1-04445 
1-04551 

52540 
52566 
52591 
52617 
52642 

1-10704 
1-10817 
1-10931 
1-11045 
1-11159 

.54083 
-54109 
•54135 
•54161 
.54187 

1-17786 
1-17909 
1-18031 
1.18154 
1.18277 

•55641 
•55667 
•55693 
•55719 
•55745 

1-25432 
1-25565 
1-25697 
1-25830 
1-25963 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

51138 
51163 
51189 
51214 
51239 

1.04658 
1-04764 
1-04870 
1-04977 
1-05084 

52668 
52694 
52719 
52745 
52771 

1.11274 
1.11388 
1-11503 
1-11617 
1.11732 

•54213 
•54238 
•54264 
54290 
54316 

1.18401 
1-18524 
1-18648 
1.18772 
1-18895 

•55771 
55797 
55823 
55849 
55876 

1.26097 
1-26230 
1.26364 
1-26498 
1.26632 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

51265 
51290 
51316 
51341 
51366 

1-05191 
1-05298 
1-05405 
1-05512 
1-05619 

52796 
52822 
52848 
52873 
52899 

1-11847 
1-11963 
1-12078 
1.12193 
1-12309 

54342 
54368 
54394 
54420 
54446 

1.19019. 

1-19144 
1-19268 
1.19393 
1.19517 

55902 
55928 
55954 
55980 
56006 

1-26766 
1-26900 
1.27035 
L-  27169 
L.  27304 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

51392 
51417 
51443 
51468 
51494 

1-05727 
1-05835 
1-05942 
1-06050 
1-06158 

52924 
52950 
52976 
53001 

53027 

1-12425 
1-12540 
1-12657 
1-12773 
.  12889 

54471 
54497 
54523 
54549 
54575 

-19642 
•19767 
•19892 
-20018 
•20143 

56032 
56058 
56084 
56111 
56137 

I-  27439 
L.  27574 
L-  27710 
L-  27845 
L.  27981 

55 
56 
57 

9 

60 

51519 

1-06267 

53053 

1-13005 

54601 

•20269 

56163 

L-  28117 

60 

292    NATURAL  VERSED  SINES  AND  EXTERNAL  SEC. 

64°  65°  66°  67° 


t 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

i 

o 

2 
3 

56163 
56189 

56215 
56241 
56267 

1.28117 
1-28253 
1-28390 
1.28526 
1-28663 

57738 
57765 
57791 
57817 
57844 

1-36620 
1-36768 
1.3691" 
1-37064 
1-37212 

59326 
.59353 
.59379 
•59406 
.59433 

1-45859 
1.46020 
1.46181 
1.46342 
1-46504 

-60927 
-60954 
-60980 
.61007 
.61034 

1-55930 
1-56106 
1-56282 
1-56458 
1-56634 

O 

2 
3 
4 

5 
6 
7 
8 
9 

56294 
56320 
56346 
56372 
56398 

1-28800 
1-28937 
1-29074 
1-29211 
1-29349 

57870 
57896 
57923 
57949 
57976 

1-37361 
1-37509 
1-37658 
1-37808 
1-37957 

.59459 
-59486 
-59512 
-59539 
-59566 

1-46665 
1-46827 
1-46989 
1-47152 
1-47314 

.61061 
-61088 
-61114 
-61141 
-61168 

1-56811 
1.56988 
1-57165 
1  -  57342 
1-57520 

6 
I 

10 

11 
12 
13 
14 

56425 
56451 
56477 
56503 
56529 

1-29487 
1-29625 
1-29763 
1-29901 
1-30040 

58002 
58028 
58055 
58081 
58108 

1-38107 
1-38256 
1-38406 
1-38556 
1.38707 

-59592 
59619 
59645 
59672 
59699 

1-47477 
1-47640 
1-47804 
1-47967 
1.48131 

-61195 
.61222 
•61248 
-61275 
-61302 

1-57698 
1-57876 
1-58054 
1-58233 
1-58412 

1O 

11 

12 
13 
14 

15 
16 
17 
18 
19 

56555 
56582 
56608 
56634 
56660 

1-30179 
1.30318 
1-30457 
1-30596 
1-30735 

58134 
58160 
58187 
58213 
58240 

1-38857 

1-39008 
1-39159 
1-39311 
1-39462 

59725 
59752 
59779 
59805 
59832 

1.48295 
1-48459 
1-48624 
1.48789 
1-48954 

-61329 
-61356 
-61383 
-61409 
-61436 

1-58591 
1-58771 
1-58950 
1-59130 

1-59311 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

56687 
56713 
56739 
56765 
56791 

1-30875 
1-31015 
1-31155 
1-31295 
1-31436 

58266 
58293 
58319 
58345 
58372 

1-39614 
1-39766 
1-39918 
1-40070 
1-40222 

59859 
59885 
59912 
59938 
59965 

1-49119 
1-49284 
1-49450 
1-49616 
1-49782 

-61463 
.61490 
.61517 
-61544 
.61570 

1-59491 
1-59672 
1-59853 
1-60035 
1-60217 

30 

21 
22 
23 
24 

25 
26 
27 
28 
29 

56818 
56844 
56870 
56896 
56923 

1-31576 
1-31717 
1-31858 
1-31999 
1-32140 

58398 
58425 
58451 
58478 
58504 

1-40375 
1-40528 
1-40681 
1-40835 
1.40988 

59992 
60018 
60045 
60072 
60098 

1-49948 
1-50115 
1-50282 
1  .  50449 
1-50617 

.61597 
.61624 
.61651 
.61678 
.61705 

1-60399 
1-60581 
1-60763 
1-60946 
1-61129 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

56949 
56975 
57001 
57028 
57054 

1-32282 
1-32424 
1-32566 
1-32708 
1-32850 

58531 
58557 
58584 
58610 
58637 

1-41142 
1-41296 
1.41450 
1.41605 
1-41760 

60125 
60152 
60178 
60205 
60232 

1-50784 
1.50952 
1-51120 
1.51289 
1-51457 

.61732 
.61759 
-61785 
-61812 
-61839 

1-61313 
1-61496 
1-61680 
1-61864 
1-62049 

30 

g 

33 
34 

35 
36 
37 
38 
39 

57080 
57106 
57133 
57159 
57185 

1-32993 
1-33135 
1-33278 
1-33422 
1-33565 

58663 
58690 
58716 
58743 
58769 

1-41914 
1.42070 
1-42225 
1.42380 
1.4253CL 

60259 
60285 
60312 
60339 
60365 

1.51626 
1.51795 
1.51965 
1-52134 
1.52304 

-61866 
-61893 
-61920 
-61947 
-61974 

1-62234 
l-b^419 
1.62604 
1.62790 
1-62976 

35 
36 
37 
38 
..39 

40 

41 
42 
43 
44 

57212 
57238 
57264 
57291 
57317 

1-33708 
1-33852 
1-33996 
1-34140 
1-34284 

58796 
58822 
58849 
58875 
58902 

1-42692 
1-42848 
1.43005 
1.43162 
1.43318. 

.60392 
-60419 
.  60445 
.  60472 
.  60499 

1  .  52474 
1.52645 
1.52815 
1.52986 
1.53157 

.62001 
-62027 
-62054 
-62081 
-62108 

1-63162 
1-63348 
1-63535 
1-63722 
1-63909 

40 
41 

42 
43 
44 

45 
46 
47 
48 
49 

57343 
57369 
•57396 
•57422 
.  57448 

1  -  34429 
1-34573 
1-34718 
1-34863 
1-35009 

58928 
58955 
58981 
59008 
59034 

1.43476 
1-43633 
1.43790 
1.43948 
1.44106 

.60526 
•60552 
.60579 
.60606 
.60633 

1-53329 
1-53500 
1-53672 
1-53845 
1-54017 

-62135 
-62162 
-62189 
-62216 
-  62243 

1-64097 
1-64285 
1  -  64473 
1-64662 
1-64851 

45 
46 
47 
48 

49 

50 

51 
52 
53 
54 

.57475 
•57501 
.57527 
•57554 
•57580 

1.35154 
1-35300 
1-35446 
1-35592 
1-35738 

-59061 
-59087 
-59114 
-59140 
.59167 

1  .  44264 
1.44423 
1-44582 
1.44741 
1-44900 

•60659 
-60686 
-60713 
.  60740 
.60768 

.54190 
.54363 
•54536 
-  54709 
-54883 

-62270 
-62297 
-62324 
-62351 
-62378 

1-65040 
1.65229 
1-65419 
1-65609 
1-65799 

50 

51 

52 
53 
54 

55 
56 
57 
58 
59 

•57606 
•57633 
.57659 
.57685 
-57712 

1-35885 
1-36031 
1-36178 
1-36325 
1-36473 

.59194 
•59220 
-59247 
.59273 
-59300 

1-45059 
1.45219 
1.45378 
1.45539 
1-45699 

.60793 
.60820 
-60847 
.60873 
.60900 

.55057 
.55231 
•55405 
.55580 

1.55755 

-62405 
-62431 
-62458 
-62485 

.62512 

1-65989 
1-66180 
1-66371 
1-66563 

1.66755 

55 
56 
57 
58 
59 

60 

•57738 

1.36620 

.59326 

1-45859 

-60927 

1-55930 

-62539 

1-66947 

60 

NATURAL  VERSED   SINES  AND  EXTERNAL  SEC.    293 

68°  69°  70°  71° 


' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

/ 

o 

1 
2 
3 

4 

.62539 
•62566 
.62593 
.62620 
•62647 

1-66947 
1-67139 
1.67332 
1.67525 
1-67718 

-64163 
•64190 
•64218 
-  64245 
•64272 

1-79043 
1-79254 
1-79466 
1-79679 
1-79891 

•65798 
•65825 
•65853 
-65880 
.65907 

1-92380 
1.92614 
1-92849 
1-93083 
1-93318 

-  67443 
•67471 
•67498 
•67526 
-67553 

2-07155 
2-07415 
2-07675 
2-07936 
2-08197 

0 
1 

2 
3 

5 
6 
7 
8 
9 

•62674 
•62701 
.62728 
.62755 
•62782 

1-67911 
1-68105 
1-68299 
1-68494 
1-68689 

-  64299 
•64326 
-64353 
-64381 
•  64408 

1-80104 
1-80318 
1-80531 
1-80746 
1-80960 

•65935 
.65962 
•65989 
-66017 
-  66044 

1.93554 
1-93790 
1-94026 
1-94263 
1-94500 

•67581 
•67608 
•67636 
•67663 
•67691 

2-08459 
2-08721 
2-08983 
2-09246 
2-09510 

5 
6 

8 

9 

10 

11 
12 
13 
14 

•62809 
•62836 
•62863 
•62890 
•62917 

1.68884 
1-69079 
1-69275 
1.69471 
1.69667 

•64435 
.  64462 
.  64489 
.64517 
.  64544 

1-81175 
1-81390 
1-81605 
1-81821 
1-82037 

-66071 
-66099 
•66126 
-66154 
.66181 

1-94737 
1-94975 
1-95213 
1-95452 
1-95691 

•67718 
-67746 
.67773 
•67801 
•67829 

2-09774 
2-10038 
2.10303 
2-10568 
2-10834 

10 

11 

12 
13 
14 

15 
16 
17 
18 
19 

•62944 
•62971 
•62998 
.63025 
.63052 

1-69864 
1-70061 
1-70258 
1.70455 
1-70653 

•64571 
.64598 
•64625 
.64653 
-64680 

1-82254 
1-82471 
1-82688 
1-82906 

1-83124 

.66208 
.66236 
•66263 
-66290 
•66318 

1-95931 
1-96171 
1-96411 
1-96652 
1-96893 

•67856 
•67884 
•67911 
.67939 
•67966 

2-11101 
2-11367 
2-11635 
2-11903 

2-12171 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

•63079 
•63106 
•63133 
•63161 
•63188 

1-70851 
1-71050 
1-71249 
1  .  71448 
1.71647 

-64707 
.  64734 
.64761 
-64789 
-64816 

1-83342 
1-83561 
1-83780 
1-83999 
1-84219 

•66345 
•66373 
-  66400 
-66427 
•  66455 

1-97135 
1-97377 
1-97619 
1-97862 
1-98106 

-67994 
•68021 
-68049 
•68077 
.68104 

2-12440 
2-12709 
2-12979 
2-13249 
2-13520 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

•63215 
-63242 
-63269 
•63296 
-63323 

1-71847 
1  .  72047 
1-72247 
1  .  72448 
1  .  72649 

-  64843 
-  64870 
-  64898 
-  64925 
-64952 

1  .  84439 
1.84659 
1.84880 
1-85102 
1-85323 

-66482 
•66510 
66537 
-66564 
-66592 

1-98349 
1-98594 
1-98838 
1.99083 
1-99329 

.68132 
•68159 
•68187 
-68214 
•  68242 

2-13791 
2-14063 
2-14335 
2-14608 
2-14881 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

•63350 
-63377 
.  63404 
.63431 
.63458 

1.72850 
1-73052 
1-73254 
1-73456 
1.73659 

-64979 
•65007 
-65034 
-65061 
-65088 

1-85545 
1-85767 
1-85990 
1-86213 
1-86437 

•66619 
-66647 
-66674 
66702 
-66729 

1-99574 
1-99821 
2-00067 
2-00315 
2-00562 

•68270 
-68297 
.68325 
68352 
.68380 

2-15155 
2-15429 
2-15704 
2-15979 
2-16255 

30 

sa 

32 
33 
34 

3!i 
36 
37 
38 
39 

.63485 
•63512 
•63539 
63566 

.63594 

1-73862 
1-74065 
1-74269 
1-74473 
1-74677 

-65116 
-65143 
-65170 
-65197 

.65225 

1-86661 
1-86885 
1-87109 
1-87334 
1.87560 

-66756 
-66784 
-66811 
.66839 
-66866 

2-00810 
2-01059 
2-01308 
2-01557 
2-01807 

•  68408 
-68435 
-68463 
-68490 
-68518 

2-16531 
2-16808 
2-17085 
2-17363 

2-17641 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

.63621 
•63648 
•63675 
•63702 
•63729 

1-74881 
1-75086 
1-75292 
1-75497 
1-75703 

-65252 
-65279 
•65306 
•65334 
-65361 

1.87785 
1.88011 
1.88238 
1-88465 
1.88692 

-66894 
-66921 
-66949 
-66976 
•  67003 

2-02057 
2-02308 
2-02559 
2-02810 
2-03062 

-68546 
•68573 
-68601 
-68628 
.68656 

2-17920 
2-18199 
2-18479 
2-18759 
2-19040 

40 

41 

42 
43 
44 

45 
46 
47 
48 
49 

•63756 
•63783 
-63810 
-63838 
.63865 

1.75909 
1-76116 
1-76323 
1-76530 
1-76737 

•65388 
•65416 
•  65443 
.65470 
•65497 

1.88920 
1-89148 
1.89376 
1.89605 
1.89834 

•67031 
•67058 
•67086 
•67113 
.67141 

2-03315 
2-03568 
2-03821 
2-04075 
2-04329 

.68684 
-68711 
.68739 
-68767 
-68794 

2-19322 
2-19604 
2-19886 
2-20169 
2-20453 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

•63892 
-63919 
•63946 
.63973 
-  64000 

1-76945 
1-77154 
1-77362 
1.77571 
1-77780 

•65525 
•65552 
•65579 
.65607 
•65634 

1-90063 
1-90293 
1.90524 
1.90754 
1-90986 

•67168 
•67196 
•67223 
•67251 
.67278 

2-04584 
2-04839 
2-05094 
2-05350 
2-05607 

•68822 
•68849 
•68877 
•68905 
.68932 

2-20737 
2-21021 
2-21306 
2-21592 
2-21878 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

-64027 
-64055 
-  64082 
-64109 
-64136 

1-77990 
1.78200 
1-78410 
1-78621 
L78832 

•65661 
-65689 
•65716 
•65743 
•65771 

1-91217 
1.91449 
1-91681 
1-91914 
1-92147 

.67306 
-67333 
•67361 
-67388 
-67416 

2-05864 
2-06121 
2-06379 
2-06637 

2-06896 

•68960 
•68988 
.69015 
.  69043 
•  69071 

2-22165 
2-22452 
2  -  22740 
2-23028 

2-23317 

55 
56 
57 
58 
59 

6O 

.64163 

1-79043 

•65798 

1-92380 

.  67443 

2-07155 

•69098 

2-23607 

60 

NATURAL  VERSED  SINES  AND  EXTERNAL  SEC. 
72°  73°  74°  75° 


t 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

' 

0 

1 

2 
3 
4 

.69098 
•69126 
.69154 
•69181 
.69209 

2-23607 
2-23897 
2-24187 
2-24478 
2-24770 

-70763 
-70791 
•70818 
-70846 
-70874 

2-42030 
2-42356 
2-42683 
2-43010 
2-43337 

-72436 
•72464 
-72492 
-72520 
-72548 

2-62796 
2-63164 
2-63533 
2-63903 
2-64274 

-74118 
-74146 
-74174 
.74202 
-74231 

2-86370 
2-86790 
2-87211 
2-87633 
2-88056 

0 

2 
3 
4 

5 
6 
7 
8 
9 

.69237 
.69264 
.69292 
.69320 
.69347 

2-25062 
2-25355 
2-25648 
2-25942 
2-23237 

70902 
70930 
70958 
70985 
71013 

2-43666 
2-43995 
2-44324 
2-44655 
2.44986 

-72576 
-72604 
-72632 
-72660 
.72688 

2-64645 
2-65018 
2-65391 
2-65765 
2-66140 

-  74259 
-74287 
-74315 
-  74343 
.74371 

2-88479 
2-88904 
2-89330 
2-89756 
2-90184 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

-69375 
.69403 
.69430 
.69458 
-69486 

2-26531 
2-26827 
2-27123 
2-27420 
2-27717 

71041 
71069 
71097 
71125 
71153 

2-45317 
2-45650 
2-45983 
2.46316 
2-46651 

.72716 
.  72744 
.72772 
-72800 
-72828 

2-66515 
2-66892 
2-67269 
2-67647 
2-68025 

-74399 
-  74427 
.  74455 
-  74484 
•74512 

2-90613 
2-91042 
2-91473 
2-91904 
2-92337 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

.69514 
•69541 
.69569 
-69597 

•69624 

2-28015 
2-28313 
2-28612 
2-28912 

2-29212 

71180 
71208 
71236 
71264 
71292 

2-46986 
2-47321 
2-47658 
2-47995 
2-48333 

72856 
72884 

72912 
72940 
72938 

2-68405 
2-68785 
2-69167 
-69549 
-69931 

.  74540 
-74568 
-74596 
-74624 
-74652 

2-92770 
2-93204 
2-93640 
2-94076 
2-94514 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

.69652 
-69680 
.69708 
.69735 
.69763 

2-29512 
2-29814 
2-30115 
2-30418 
2.30721 

71320 
71348 
71375 
71403 
71431 

2-48671 
2-49010 
2-49350 
2-49691 
2-50032 

72996 
73024 
73052 
73080 
73108 

-70315 
-70700 
-71085 
-71471 
-71858 

-74680 
.74709 
-74737 
-74765 
-74793 

2-94952 
2-95392 
2-95832 
2-96274 
2-96716 

20 

21 

22 
23 
24 

25 
26 
27 
28 
29 

•69791 
•69818 
-69846 
-69874 
-69902 

2-31024 
2-31328 
2-31633 
2-31939 
2-32244 

71459 
71487 
71515 
71543 
71571 

2-50374 
2-50716 
2-51060 
2-51404 
2-51748 

73136 
73164 
73192 
73220 
73248 

-  72246 
-72635 
-73024 
-73414 
-73806 

-74821 
-74849 
-74878 
-74906 
-74934 

2-97160 
2-97604 
2-98050 
2-98497 
2-98944 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

-69929 
.69957 
.69985 
.70013 
.  70040 

2-32551 
2-32858 
2-33166 
2-33474 
2-33783 

71598 
71626 
71654 
71682 
71710 

2-52094 
2-52440 
2-52787 
2-53134 
2-53482 

73276 
73304 
73332 
73360 
73388 

-74198 
-74591 
-74984 
-75379 
-75775 

.74962 
74990 
75018 
-75047 
-75075 

2-99393 
2.99843 
3-00293 
3  •  00745 
3-01198 

30 

31 
32 
33 
34 

35 
36 
37 
38 
31- 

•70068 
.70096 
-70124 
.70151 
-70179 

2.34092 
2-34403 
2-34713 
2-35025 
2-35336 

71738 
71766 
71794 
71822 
71850. 

2-53831 
2-54181 
2-54531 
2-54883 

2-55235 

73416 
73444 
73472 
73500 

73529  . 

-76171 
-76568 
-76966 
2-77365 

2-77765 

-75103 
-75131 
-75159 
-75187 
.75216 

3-01652 
3-02107 
3-02563 
3-03020 

3-03479 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

.70207 
-70235 
.  70263 
.70290 
-70318 

2-35649 
2-35982 
2-36276 
2-36590 
2-36905 

71877 
71905 
71933 
71961 
71989 

2-55587 
2-55940 
2-56294 
2-56649 
2-57005 

73557 
73585 
73613 
-73641 
-73669 

2-78166 
2-78568 
2-78970 
2-79374 
2-79778 

-75244 
-75272 
-75300 
-75328 
.75356 

3-03938 
3-04398 
3-04860 
3-05322 
3-05786 

4O 

41 

42 
43 
44 

45 
46 
47 
48 
49 

•  70346 
•70374 
.  70401 
•70429 
.  70457 

2-37221 
2-37537 
2-37854 
2-38171 
2-38489 

72017 
72045 
72073 
-72101 
.72129 

2-57361 
2-57718 
2-58076 
2-58434 
2-58794 

-73697 
•73725 
.73753 
-73781 
•73809 

2-80183 
2-80589 
2-80996 
2-81404 
2-81813 

•75385 
.75413 
-75441 
-75469 
-75497 

3-06251 
3-06717 
3-07184 
3-07652 
3-08121 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

•70485 
.70513 
•70540 
.70568 
.70596 

2-38808 
2-39128 
2-39448 
2-39768 
2.40089 

.72157 
•72185 
•72213 
.72241 
•72269 

2-59154 
2-59514 
2-59876 
2-60238 
2-60601 

•73837 
.73865 
-73893 
-73921 
-73950 

2-82223 
2-82633 
2-83045 
2-83457 
2-83871 

-75526 
-75554 
•75582 
.75610 
-75fi39 

3-08591 
3-09063 
3-09535 
3-10009 
3-10484 

5O 

51 
52 
53 
54 

55 
56 
57 
58 
59_ 

.70624 
•70652 
•70679 
.70707 
•70735 

2-40411 
2-40734 
2-41057 
2-41381 
2-41705 

.72296 
•72324 
•72352 
.72380 
.  72408 

2-60965 
2-61330 
2-61695 
2-62061 

2-62428 

-73978 
-  74006 
.74034 
-74062 
-74090 

2-84285 
2-84700 
2-85116 
2-85533 
2-85951 

-75667 
•75695 
-75723 
-75751 
-75780 

3-10960 
3-11437 
3-11915 
3-12394 
3-12875 

55 
56 
57 
58 
59 

60 

•70763 

2-42030 

•72436 

2-62796 

•74118 

2-86370 

.75808 

3-13357 

6O 

NATURAL  VERSED  SINES  AND  EXTERNAL  SEC.    295 
76°  77°  78°  79° 


' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

' 

0 

1 

2 
3 
4 

75808 
•75836 
75864 
•75892 
.75921 

3-13357 
3-13839 
3-14323 
3-14809 
3-15295 

77505 
77533 
77562 
77590 
77618 

3-44541 
3-45102 
3-45664 
3-46228 
3-46793 

79209  ' 
79237  i 
79266 
79294 
79323  1 

3-80973 
3-81633 
3-82294 
3-82956 
3.83621 

.80919 
-80948 
-80976 
-81005 
•81033 

4-24084 
4-24870 
4-25658 
4-26448 
4-27241 

0 

1 
2 
3 

5 
6 
7 
8 
9 

•75949 
•75977 
•76005 
-76034 
•76062 

3-15782 
3-16271 
3-16761 
3-17252 
3-17744 

77647 
77675 
77703 
77732 
77760 

3-47360 
3-47928 
3-48498 
3.49069 
3-49642 

79351 
79380 
79408 
79437 
79465 

3  •  84288 
3-84956 
3-85627 
3-86299 
3-86973 

•81062 
-81090 
•81119 
•81148 
.81176 

4-28036 
4-28833 
4-29634 
4-30436 
4-31241 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

.76090 
•76118 
.76147 
.76175 
.  76203 

3-18238 
3-18733 
3-19228 
3-19725 
3-20224 

77788 
77817 
77845 
77874 
77902 

3-50216 
3-50791 
3-51368 
3-51947 
3-52527 

-  79493 
.79522 
.79550 
.79579 
-79607 

3-87649 
3-88327 
3-89007 
3-89689 
3-90373 

.81205 
.81233 
.81262 
-81290 
•81319 

4-32049 
4-32859 
4-33671 
4-34486 
4-35304 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

•76231 
.76260 
.76288 
•76316 

.76344 

3-20723 
3-21224 
3-21726 
3-22229 
3-22734 

77930 
77959 
.77987 
•78015 
78044 

3-53109 
3-53692 
3-54277 
3  -  54863 

3-55451 

.79636 
-79664 
-  79693 
•79721 
.79750 

3-91058 
3-91746 
3-92436 
3-93128 

3-93821 

.81348 
•81376 
-81405 
•81433 

.81462 

4-36124 
4-36947 
4-37772 
4-38600 
4  39430 

15 
16 
17 
18 
-12 

20 

21 
22 
23 
24 

.76373 
.76401 
-  76429 
.76458 
-76486 

3-23239 
3-23746 
3-24255 
3-24764 
3-25275 

.78072 
.78101 
.78129 
.78157 
•78186 

3-56041 
3-56632 
3-57224 
3-57819 
3-58414 

.79778 
•79807 
-79835 
-79864 
.  79892 

3-94517 
3-95215 
3-95914 
3-96616 
3-97320 

.81491 
•81519 
.81548 
•81576 
•81605 

4-40263 
4-41099 
4-41937 
4-42778 
4-43622 

20 

2. 
2* 
23 
24 

25 
26 
27 
28 
29 

.76514 
•76542 
.76571 
•76599 
.76627 

3-25787 
3-  26300 
3-26814 
3-27330 
3-27847 

.78214 
.  78242 
.78271 
-78299 
.78328 

3-59012 
3-59611 
3-60211 
3-60813 
3-61417 

-79921 
-  79949 
-79978 
-80006 
-80035 

3-98025 
3-98733 
3-99443 
4-00155 
4-00869 

.81633 
•81662 
•81691 
.81719 
•81748 

4-44468 
4-45317 
4-46169 
4-47023 
4-47881 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

•76655 
•76684 
•76712 
•  76740 
76769 

3-28366 
3-28885 
3-29406 
3-29929 
3.30452 

-78356 
-78384 
•78413 
-  78441 
-  78470 

3  -  62023 
3-62630 
3-63238 
3-63849 
3  -  64461 

•80063 
-80092 
.80120 
•80149 
•80177 

4-01585 
4-02303 
4-03024 
4-03746 
4-04471 

•81776 
•81805 
•81834 
•81862 
•81891 

4-48740 
4-49603 
4-50468 
4-51337 
4-52208 

3O 

31 
32 
33 
-34 

35 
36 
37 
38 
39 

•76797 
•76825 
•76854 
•76882 
-76910 

3-30977 
3-31503 
3-32031 
3-32560 
3-33090 

.  78498 
•78526 
-78555 
-78583 
.78612 

3-65074 
3-65690 
3-66307 
3-66925 

3-67545 

•80206 
-80234 
.80263 
-80291 
-80320 

4-05197 
4-05926 
4-06657 
4-07390 
4-08125 

•81919 
.81948 
.81977 
.82005 
•82034 

4-53081 
4-53958 
4-54837 
4-55720 

4.56605 

35 
38 
37 
38 
-39. 

40 

41 
42 
43 
44 

•76938 
•76967 
•76995 
•  77023 
•77052 

3-33622 
3-34154 
3-34689 
3-35224 
3-35761 

.  78640 
•78669 
-78697 
•78725 
.  78754 

3-68167 
3-68791 
3.  694]  7 
3  -  70044 
3  -  70673 

-80348 
-80377 
-80405 
-  80434 
-80462 

4-08863 
4-09602 
4-10344 
4-11088 
4-11835 

•82063 
•82091 
•82120 
.82148 
-82177 

4-57493 
4-58383 
4-59277 
4-60174 
4-61073 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

•77080 
•77108 
•77137 
•77165 
•77193 

3-36299 
3-36839 
3-37380 
3-37923 
3-38466 

.78782 
.78811 
.78839 
.78868 
•78896 

3-71303 
3-71935 
3-72569 
3-73205 
3-73843 

-8049' 
-80520 
-80548 
-80577 
-80605 

4-12583 
4-13334 
4-14087 
4-14842 
4-15599 

-82206 
-82234 
.82263 
•82292 
•82320 

4-61976 
•62881 
•63790 
•64701 
.65616 

45 
48 
47 
48 
_49 

50 

51 
52 
53 
54 

•77222 
•77250 
-77278 
•77307 
•77335 

3-39012 
3-39558 
3-40106 
3-40656 
3-41206 

-78924 
-78953 
-78981 
•79010 
•79038 

3  -  74482 
3-75123 
3-75766 
3-76411 
3-77057 

•80634 
.80662 
.80691 
.80719 
.  80748 

4-16359 
4-17121 
4-17886 
4-18652 
4-19421 

•82349 
•82377 
.82406 
.82435 
..82463 

-66533 
-67454 
-68377 
-69304 
-70234 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

•77363 
•77392 
.  77420 
-  77448 

•77477 

3-41759 
3-42312 
3-42867 
3-43424 
3-43982 

.79067 
•79095 
•79123 
•79152 
•79180 

3-77705 
3-78355 
3-79007 
3-79661 
3.80316 

•80776 

.80805 
-80833 
-80862 
•80891 

4-20193 
4-20966 
4-21742 
4-22521 
4-23301 

.82492 
.82521 
.82549 
•82578 
.82607 

.71166 
.72102 
.73041 
-73983 
4-74929 

55 
58 
57 
58 
-5§ 

60 

•77505 

8-44541 

.79209 

3-80973 

.80919 

4-24084 

.82635 

4-75877 

6O 

296    NATURAL  VERSED   SINES  AND  EXTERNAL  SEC. 

80°  81°  83°  83° 


/ 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

i 

0 

1 

2 
3 
JL 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 

.82635- 
.82664 
.82692 
-82721 
.82750 

4-75877 
4.76829 
4-77784 
4.78742 
4-79703 

•84357 
.84385 
•  84414 
.  34443 
-  84471 

5-39245 
5-40422 
5-41602 
5-42787 
5-43977 

-86083 
.86112 
.86140 
•86169 
.86188 

6.18530 
6-20020 
6-21517 
6-23019 
6-24529 

•87813 
•87842 
•87871 
•87900 
•87929 

7-20551 
7-22500 
7-24457 
7-26425 
7-28402 

0 

1 
2 
3 

4 

-82778 
-82807 
.82836 
-82864 
.82893 

4-80667 
4-81635 
4-82606 
4-83581 
4-84558 

.  84500 
•84529 
•84558 
-84586 
-84615 

5-45171 
5-46369 
5-47572 
5-48779 
5-49991 

•86227 
•86256 
.86284 
-86313 
•86342 

6-26044 
6-27566 
6-29095 
6-30630 
6-32171 

•87957 
•87986 
•88015 
•88044 
-88073 

7-30388 
7-32384 
7-34390 
7-36405 
7-38431 

5 
6 
7 
8 
__1 
10 
11 
12 
13 
14 

-82922 
-82950 
•82979 
-83008 
-83036 

4-85539 
4-86524 
4-87511 
4-88502 
4-89497 

.  84644 
•84673 
•84701 
.84730 
.84759 

5-51208 
5-52429 
5-53655 
5-54886 
5-56121 

•86371 
-86400 
-86428 
-86457 
-86486 

6-33719 
6-35274 
6-36835 
6-38403 
6-39978 

-88102 
-88131 
.88160 
-88188 
-88217 

7-40466 
7-42511 
7-44566 
7-46632 
7-48707 

-83065 
83094 
83122 
83151 
83180 

4-90495 
4.91496 
4-92501 
4-93509 
4-94521 

•84788 
.84816 
.84845 
-84874 
.  84903 

5-57361 
5-58606 
5-59855 
5-61110 
5-62369 

•86515 
•86544 
-86573 
-86601 
.86630 

6-41560 
6-43148 
6  -  44743 
6-46346 
6-47955 

-88246 
-88275 
-88304 
-88333 
-88362 

7-50793 
7-52889 
7-54996 
7-57113 
7-59241 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 

83208 
83237 
83266 
83294 
83323 

4-95536 
4-96555 
4-97577 
4-98603 
4-99633 

•84931 
-84960 
-84989 
-85018 
85046 

5-63633 
5-64902 
5-66176 
5-67454 
5-68738 

.86659 
-86688 
.86717 
-86746 
-86774 

6-49571 
6-51194 
6-52825 
6-54462 
6-56107 

-88391 
-88420 
-88448 
-88477 
.88506 

7-61379 
7-63528 
7-65688 
7-67859 
7-70041 

20 

21 
22 
23 
24 
25 
26 
27 
28 
29 

83352 
83380 
83409 
83438 
83467 

5-00666 
5-01703 
5-02743 
5-03787 
5-04834 

85075 
85104 
85133 
85162 
85190 

5-70027 
5-71321 
5-72620 
5-73924 
5-75233 

-86803 
-86832 
-86861 
-86890 
-86919 

6-57759 
6-59418 
6-61085 
6-62759 
6  -  64441 

-88535 
-88564 
-88593 
-88622 
-88651 

7-72234 
7-74438 
7-75653 
7-78880 
7-81118 

83495 
83524 
83553 
83581 
83610 

5-05886 

5-06941 
5-08000 
5-09062 
5-10129 

85219 
85248 
85277 
85305 
85334 

5-76547 
5-77866 
5-79191 
5-80521 
5-81856 

-86947 
-86976 
-87005 
-87034 
-87063 

6-66130 
6-67826 
6.69530 
6-71242 
6-72962 

-88680 
-88709 
-88737 
-88766 
-88795 

7-83367 
7-85628 
7-87901 
7-90186 
7-92482 

30 

31 
32 
33 
_3J 
35 
36 
37 
38 
-39 
40 
41 
42 
43 
44 

35 
36 
37 
38 
39 

83639 
83667 
83696 
83725 
83754 

5-11199 
5.12273 
5-13350 
5-14432 
5-15517 

85363 
85392 
85420 
85449 
85478 

5-83196 
5-84542 
5-85893 
5.87250 
5-88612 

.87092 
•87120 
.87149 
.87178 

-87207 

6-74689 
6-76424 
6-78167 
6-79918 
6-81677 

•88824 
•88853 
•88882 
-88911 
-88940 

7-94791 
7-97111 
7-99444 
8-01788 
8-04146 

40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 

55 
56 
57 
58 
.60. 
60 

83782 
83811 
83840 
83868 
83897 

5-16607 
5-17700 
5-18797 
5-19898 
5-21004 

85507 
85536 
85564 
85593 
85622 

5-89979 
5-91352 
5-92731 
5-94115 
5-95505 

-87236 
-87265 
-87294 
-87322 
-87351 

6-83443 
6-85218 
6.87001 
6.88792 
6.90592 

-88969 
.88998 
-89027 
-89055 
-89084 

8-06515 
8-08897 
8.11292 
8-13699 
8-16120 

83926 
83954 
83983 
84012 
84041 

5.22113 
5-23226 
5-24343 
5.25464 
5.26590 

85651 
85680 
85708 
•85737 
.85766 

5-96900 
5-98301 
5-99708 
6-01120 
6-02538 

-87380 
-  87409 
-87438 
-87467 
-87496 

6-92400 
6-94216 
6-96040 
6-97873 
6-99714 

-89113 
-89142 
•89171 
.89200 
.89229 

8-18553 
8-20999 
8-23459 
8-25931 
8-28417 

45 
46 
47 
48 
49 

84069 
-84098 
-84127 
.84155 
.84184 

5.27719 
5-28853 
5-29991 
5.31133 
5.32279 

.85795 
.85823 
•85852 
.85881 
.85910 

6-03962 
6-05392 
6-06828 
6-08269 
6-09717 

-87524 
-87553 
-87582 
-87611 
-87640 

7-01565 
7-03423 
7-05291 
7.07167 
7-09052 

•89258 
•89287 
-89316 
-89345 
-89374 

8-30917 
8-33430 
8-35957 
8-38497 
8-41052 

50 

51 
52 
53 
54 
55 
56 
57 
58 
59 

.84213 
•84242 
.  84270 
.84299 
•84328 

5-33429 
5-34584 
5-35743 
5-36906 
5-38073 

•85939 
.85967 
•85996 
.86025 
.86054 

6-11171 
6-12630 
6-14096 
6-15568 

6.17046 

-87669 
-87698 
-87726 
-87755 
-87784 

7-10946 
7.12849 
7-14760 
7-16681 

7-18612 

.89403 
•89431 
•89460 
•89489 
-89518 

8-43620 
8-46203 
8-48800 
8-51411 
8.54037 

•84357 

5-39245 

.86083 

6.18530 

.87813 

7.20551 

.89547 

8-56677 

60 

NATURAL  VERSED  SINES   AND  EXTERNAL  SEC.    297 

84°  85°  86° 


' 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

i 

0 

1 

2 
3 

.89547 
.89576 
•89605 
.89634 
.89663 

8.56677 
8.59332 
8.62002 
8.64687 
8.67387 

.91284 
.91313 
.91342 
•91371 
-91400 

10.47371 
10-51199 
10.55052 
10.58932 
10-82337 

-93024 
.93053 
.93082 
-93111 
.93140 

13-33559 
13-39547 
13-45586 
13-51676 
13-57817 

0 

1 

2 
3 

4 

5 
6 
7 
8 
9 

•89692 
•89721 
•89750 
•89779 
•89808 

8-70103 
8-72833 
8-75579 
8-78341 
8-81119 

.91429 
-91458 
-91487 
-91516 
.91545 

10-66769 
10-70728 
10-74714 
10-78727 
10-82768 

.93169 
•93198 
•93227 
•93257 
•93286 

13-64011 
13-70258 
13-76558 
13-82913 
13-89323 

5 
6 
7 
8 
9 

10 

11 
12 
13 
14 

.89836 
•89865 
.89894 
•89923 
.89952 

8-83912 
8-86722 
8.89547 
8-92389 
8-95248 

•91574 
-91603 
.91632 
.91661 
-91690 

10-88837 
10-90934 
10-95060 
10.99214 
11-03397 

•93315 
•93344 
•93373 
.  93402 
.93431 

13-95788 
14-02310 
14-08890 
14-15527 
14-22223 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19 

•89981 
•90010 
•90039 
•90068 
•90097 

8-98123 
9.01015 
9-03923 
9-06849 
9-09792 

-91719 
-91748 
-91777 
-91806 
•91835 

1-07610 
11-11852 
11-16125 
11.20427 
11-24761 

•93460 
•93489 
•93518 
•93547 
-93576 

14-28979 
14-35795 
14-42672 
14-49611 
14-56614 

15 
16 
17 
18 
.19 

20 

21 
22 
23 
24 

•90126 
•90155 
•90184 
•90213 
•90242 

9.12752 
9-15730 
9.18725 
9-21739 
9-24770 

•91864 
•91893 
•91922 
•91951 
•91980 

11-29125 
11.33521 
11.37948 
11.42408 
11.46900 

•93605 
•93634 
•93663 
•93692 
•93721 

14-63679 
14-70810 
14-78005 
14-85268 
14-92597 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

•90271 
•90300 
.90329 
•90358 
•90386 

9-27819 
9-30887 
9-33973 
9.37077 
9.40201 

•92009 
.92038 
•92067 
-92096 
-92125 

11.51424 
11-55982 
11-60572 
11.65197 
11.69856 

•93750 
•93779 
•93808 
•93837 
•93866 

14-99995 
15-07462 
15-14999 
15-22607 
15-30287 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

•90415 
•  90444 
•  90473 
•90502 
•90531 

9-43343 
9-46505 
9-49685 
9-52886 
9-56106 

.92154 
•92183 
.92212 
.92241 
-92270 

11-74550 
n.  79278 
11-84042 
11-88841 
11.93677 

•93895 
•93924 
•93953 
•93982 
•94011 

15-38041 
15-4b869 
15-53772 
15-61751 
15-69808 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39 

•90560 
•90589 
•90618 
•90647 
•90676 

9-59346 
9-62605 
9-65885 
9-69186 
9-72507 

.92299 
.92328 
-92357 
-92386 
•92415 

11-98549 
12-03458 
12-08404 
12.13388 
12.18411 

•  94040 
•94069 
•94098 
•94127 
•94156 

15-77944 
15-86159 
15-94456 
16-02835 
16-11297 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

•90705 
•90734 
•90763 
•90792 
-90821 

9-75849 
9-79212 
9-82596 
9-86001 
9.89428 

•  92444 
•92473 
-92502 
-92531 
-92560 

12-23472 
12-28572 
12-33712 
12.38891 
12-44112 

•94186 
•94215 
•  94244 
•94273 
•94302 

16-19843 
16-28476 
16.37196 
16-46005 
16-54903 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

•90850 
•90879 
•90908 
•90937 
•90966 

9-92877 
9-96348 
9-99841 
10-03356 
10-06894 

.92589 
-92618 
-92647 
-92676 
-92705 

12-49373 
12-54676 
12-60021 
12.65408 
12-70838 

•94331 
•94360 
-94389 
•94418 
-  94447 

16-63893 
16-72975 
16-82152 
16-91424 
17-00794 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

•90995 
•91024 
.91053 
.91082 
.91111 

10-10455 
10-14039 
10.17646 
10-21277 
10-24932 

•92734 
-92763 
.92792 
.92821 
-92850 

12-76312 
12-81829 
12-87391 
12-92999 
12-98651 

•  94476 
•94505 
•94534 
•94563 
•94592 

17-10262 
17-19830 
17-29501 
17-39274 
17-49153 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59 

•91140 
•91169 
.91197 
•91226 
•91255 

10-28610 
10.32313 
10-36040 
10-39792 
10-43569 

-92879 
-92908 
.92937 
-92966 
•92995 

13-04350 
13-10096 
13.15889 
13-21730 
13-27620 

•94621 
•94650 
•94679 
•94708 
.94737 

17-59139 
17-69233 
17-79438 
17-89755 
18-00185 

55 
56 
57 
58 
-59. 

60 

.91284 

10.47371 

.93024 

13-33559 

•94766 

18.10732 

60 

NATURAL  VERSED  SINES   AND  EXTERNAL  SEC. 
87°  88°  89° 


• 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

Vers. 

Ex.  sec. 

> 

o 

2 
3 

.94766 
•94795 
.94825 
.94854 
.94883 

18-10732 
18-21397 
18-32182 
18-43088 
18-54119 

.96510 
.96539 
.96568 
•96597 
-96626 

27-65371 
27-89440 
28-13917 
28-38812 
28-64137 

•98255 
•98284 
•98313 
-98342 
-98371 

53-29869 
57.26976 
58-27431 
59-31411 
60-39105 

0 

2 
3 
4 

5 
6 
7 
8 
9 

•94912 
•94941 
.94970 
.94999 
.95028 

18-65275 
18-76560 
18-87976 
18-99524 
19-11208 

.96655 
.96684 
.96714 
.96743 
.96772 

28-89903 
29-16120 
29-42802 
29-69960 
29.97607 

•98400 
•98429 
.98458 
.98487 
.98517 

61.50715 
62.66460 
63-86572 
65-11304 
66-40927 

5 
6 

8 
9 

10 

11 
12 
13 
14 

.95057 
.95086 
•95115 
.95144 
.95173 

19.23028 
19-34989 
19-47093 
19-59341 
19-71737 

.96801 
.96830 
-96859 
-96888 
-96917 

30-25758 
30-54425 
30-83623 
31-13366 
31-43671 

•98546 
•98575 
-98604 
.98633 
.98662 

67-75736 
69-16047 
70-62207 
72-14583 
73-73586 

10 

11 
12 
13 
14 

15 
16 
17 
18 
19_ 

-95202 
•95231 
.95260 
.95289 
.95318 

19-84283 
19-96982 
20-09838 
20-22852 
20-36027 

-96946 
-96975 
.97004 
.97033 
.97062 

31-74554 
32.06030 
32-38118 
32-70835 
33-04199 

•98691 
•98720 
•98749 
.98778 
.98807 

75-39655 
77-13274 
78-94968 
80-85315 
82-84947 

15 
16 
17 
18 
19 

20 

21 
22 
23 
24 

.95347 
.95377 
.95406 
.95435 
.95464 

20-49368 
20-62876 
20-76555 
20-90409 
21  .  04440 

.97092 
.97121 
.97150 
.97179 
.97208 

33-38232 
33-72952 
34-08380 
34-44539 
34-81452 

.98836 
•98866 
.98895 
.98924 
•98953 

84-94561 
87-14924 
89-46886 
91-91387 
94-49471 

20 

21 
22 
23 
24 

25 
26 
27 
28 
29 

.95493 
•95522 
•95551 
.95580 
.95609 

21-18653 
21.33050 
21-47635 
21-62413 
21-77386 

•97237 
.97266 
.97295 
.97324 
•97353 

35-19141 
35-57633 
35-96953 
36-37127 
36-78185 

.98982 
.99011 
.99040 
-99069 
-99098 

97-22303 
100-1119 
103-1757 
106-4311 
109-8966 

25 
26 
27 
28 
29 

30 

31 
32 
33 
34 

.95638 
.95667 
•95696 
.95725 
•95754 

21-92559 
22-07935 
22-23520 
22-39316 
22-55328 

.97382 
.97411 
.97440 
.97470 
.97499 

37-20155 
37-63068 
38-06957 
38-51855 
38-97797 

.99127 
.99156 
-99186 
.99215 
.99244 

113-5930 
117-5444 
121-7780 
126-3253 
131-2223 

30 

31 
32 
33 
34 

35 
36 
37 
38 
39_ 

•95783 
.95812 
•95842 
•95871 
•95900 

22.71563 
22-88022 
23-04712 
23-21637 
23-38802 

.97528 
.97557 
.97586 
-97615 
.97644 

39-44820 
39-92963 
40-42266 
40-92772 
41-44525 

.99278 
.99302 
.99331 
.99360 
•99389 

136-5111 
142-2406 
148-4684 
155-2623 
162-7033 

35 
36 
37 
38 
39 

40 

41 
42 
43 
44 

•95929 
•95958 
•95987 
.96016 
•96045 

23-56212 
23-73873 
23-91790 
24-09969 
24-28414 

•97673 
.97702 
.97731 
.97760 
.97789 

41-97571 
42-51961 
43-07746 
43-64980 
44-23720 

-99418 
-99447 
.99476 
.99505 
.99535 

170-8883 
179-9350 
189-9868 
201-2212 
213-8600 

40 

41 
42 
43 
44 

45 
46 
47 
48 
49 

•96074 
.96103 
.96132 
.96161 
.96190 

24-47134 
24-66132 
24-85417 
25-04994 
25-24869 

•97819 
•97848 
•97877 
.97906 
.97935 

44-84026 
45-45963 
46-09596 
46-74997 
47-42241 

•99564 
.99593 
.99622 
.99651 
.99680 

228-1839 
244.5540 
263-4427 
285-4795 
311.5230 

45 
46 
47 
48 
49 

50 

51 
52 
53 
54 

.96219 
.96248 
.96277 
.96307 
.96336 

25-45051 
25-65546 
25-86360 
26.07503 
26-28981 

•97964 
-97993 
-98022 
-98051 
.98080 

48-11406 
48-82576 
49-55840 
50.31290 
51.09027 

.99709 
.99738 
.99767 
.99796 
.99825 

342.7752 
380-9723 
428-7187 
490-1070 
571-9581 

50 

51 
52 
53 
54 

55 
56 
57 
58 
59_ 

.96365 
.96394 
.96423 
.96452 
.96481 

26-50804 
26-72978 
26-95513 
27-18417 
27-41700 

.98109 
-98138 
-98168 
.98197 
•98226 

51-89156 
52-71790 
53-57048 
54-45053 
55-35946 

.99855 
.99884 
.99913 
•99942 
•99971 

686-5496 
858-4369 
1144-916 
1717-874 
3436-747 

55 
58 
57 
58 
59 

60 

.96510 

27-65371 

.98255 

56-29869 

1-00000 

Infinite 

60 

TABLES 


299 


RADII  —  METRIC 
R      "        » 


Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

0°00' 

00 

oo 

1°00' 

1145.930 

3.05916 

2°  00' 

572.987 

2.75814 

01 

68754.931 

4.83730 

01 

1127.145 

3.05198 

01 

568.252 

2.75454 

02 

34377.468 

4.53627 

02 

1108.966 

3.04492 

02 

563.595 

2.75097 

03 

22918.310 

4.36018 

03 

1091.364 

3.03797 

03 

559.013 

2.74742 

04 

17188.735 

4.23524 

04 

1074.311 

3.03113 

04 

554.505 

2.74391 

05 

13750.987 

4.13833 

05 

1057.785 

3.02440 

05 

550.070 

2.74042 

06 

11459.157 

4.05912 

06 

1041.757 

3.01777 

06 

545.705 

2.73696 

07 

9822.134 

3.99221 

07 

1026.209 

3.01124 

07 

541.408 

2.73352 

08 

8594.369 

3.93421 

08 

1011.118 

3.00480 

08 

537.179 

2.73012 

09 

7639.438 

3.88306 

09 

996.465 

2.99846 

09 

533.015 

2.72674 

10 

6875.496 

3.83730 

10 

982.230 

2.99221 

10 

528.916 

2.72339 

11 

6250.451 

3.79591 

11 

968.396 

2.98605 

11 

524.879 

2.72006 

12 

5729.582 

3.75812 

12 

954.947 

2.97998 

12 

520.903 

2.71676 

13 

5288.845 

3.72336 

13 

941.866 

2.97399 

13 

516.985 

2.71348 

14 

4911.070 

3.69118 

14 

929.139 

2.96808 

14 

513.129 

2.71023 

15 

4583.666 

3.66121 

15 

916.751 

2.96225 

15 

509.328 

2.70700 

16 

4297.187 

3.63318 

16 

904.689 

2.95650 

16 

505.584 

2.70379 

17 

4044.412 

3.60686 

17 

892.940 

2.95082 

17 

501.894 

2.70061 

18 

3819.723 

3.58203 

18 

881.492 

2.94522 

18 

498.258 

2.69745 

19 

3618.685 

3.55855 

19 

870.335 

2.93969 

19 

494.674 

2.69432 

20 

3437.752 

3.53627 

20 

859.456 

2.93422 

20 

491.141 

2.69121 

21 

3274.050 

3.51509 

21 

848.846 

2.92883 

21 

487.658 

2.68812 

22   3125.230 

3.49488 

22 

838.495 

2.92350 

22 

484.224 

2.68505 

23   2989.351 

3.47558 

23 

828.393 

2.91824 

23 

480.838 

2.68200 

24 

2864.795 

3.45709 

24 

818.532 

2.91304 

24 

477.500 

2.67897 

25 

2750.203 

3.43936 

25 

808.902 

2.90790 

25 

474.207 

2.67597 

26 

2644.427 

3.42233 

26 

799.497 

2.90282 

26 

470.960 

2.67298 

27 

2546.485 

3.40594 

27 

790.308 

2.89780 

27 

467.756 

2.67002 

28 

2455.540 

3.39015 

28 

781.327 

2.89283 

28 

464.596 

2.66708 

29 

2370.867 

3.37491 

29 

772.549 

2.88793 

29 

461.479 

2.66415 

30 

2291.838 

3.36018 

30 

763.966 

2.88307 

30 

458.403 

2.66125 

31   2217.909 

3.34594 

31 

755.571 

2.87828 

31 

455.367 

2.65836 

32  '  2148.600 

3.33216 

32 

747.359 

2.87353 

32 

452.372 

2.65550 

33   2083.491 

3.31879 

33 

739.323 

2.86883 

33 

449.416 

2.65265 

34  !  2022.212 

3.30583 

34 

731.458 

2.86419 

34 

446.498 

2.64982 

35  1  1964.435 

3.29324 

35 

723.759 

2.85959 

35 

443.618 

2.64701 

36  |  1909.868 

3.28100 

36 

716.221 

2.85505 

36 

440.775 

2.64422 

37  ;  1858.251 

3.26910 

37 

708.837 

2.85055 

37 

437.968 

2.64144 

38  !  1809.350 

3.25752 

38 

701.605 

2.84609 

38 

435.196 

2.63868 

39   1762.957 

3.24624 

39 

694.518 

2.84168 

39 

432.460 

2.63595 

40 

1718.883 

3.23525 

40 

687.574 

2.83732 

40 

429.757 

2.63322 

41 

1676.959 

3.22452 

41 

680.766 

2.83300 

41 

427.088 

2.63052 

42 

1637.032 

3.21406 

42 

674.093 

2.82872 

42 

424.452 

2.62783 

43 

1598.962 

3.20384 

43 

667.549 

2.82448 

43 

421.849 

2.62516 

44 

1562.623 

3.19385 

44 

661  .  130 

2.82029 

44 

419.277 

2.62250 

45 

1527.899 

3.18409 

45 

654.834 

2.81613 

45 

416.737 

2.61986 

46 

1494.684 

3.17455 

46 

648.657 

2.81202 

46 

414.227 

2.61724 

47 

1462.883 

3.16521 

47 

642.595 

2.80794 

47 

411.747 

2.61463 

48 

1432.406 

3.15607 

48 

636.646 

2.80390 

48 

409.296 

2.61204 

49 

1403.174 

3.14711 

49 

630.806 

2.79990 

49 

406.875 

2.60946 

50 

1375.111 

3.13834 

50 

625.072 

2.79593 

50 

404.482 

2.60690 

51 

1348.148 

3.12974 

51 

619.441 

2.79200 

51 

402.117 

2.60435 

52 

1322.223 

3.12130 

52 

613.910 

2.78810 

52 

399.780 

2.60182 

53 

1297.276 

3.11303 

53 

608.478 

2.78424 

53 

397.469 

2.59930 

54 

1273.253 

3.10491 

54 

603.141 

2.78042 

54 

395.186 

2.59680 

55 

1250.103 

3.09695 

55 

597.897 

2.77663 

55 

392.928 

2.59431 

56 

1227.780 

3.08912 

56 

592.743 

2.77287 

56 

390.696 

2.59184 

57 

1206.241 

3.08143 

57 

587.677 

2.76914 

57 

388.489 

2.58938 

58 

1185.444 

3.07388 

58 

582.698 

2.76544 

58 

386.307 

2.58693 

59 

1165.352 

3.06646 

59 

577.802 

2.76178 

59 

384.149 

2.58450 

300 


THE   RAILROAD   TAPER 


RAJDII  —  METRIC  —  (Continued) 


Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

3°  00' 

382.016 

2.58208 

4°  00' 

286.537 

2.45718  i 

5°  00' 

229.256 

2.36032 

[01 

379.906 

2.57968 

01 

285.348 

2.45537 

01 

228.495 

2.35888 

02 

377.818 

2.57728 

02 

284.170 

2.45358 

02 

227.739 

2.35744 

03 

375.754 

2.57490 

03 

283.001 

2.45179 

03 

226.987 

2.35600 

04 

373.713 

2.57254 

04 

281.842 

2.45001 

04 

226.241 

2.35457 

05 

371.693 

2.57018 

05 

280.692 

2.44823 

05 

225.500 

2.35315 

06 

369.695 

2.56784 

06 

279.551 

2.44646 

06 

224.764 

2.35173 

07 

367.719 

2.56552 

07 

278.420 

3.44470 

07 

224.032 

2.35031 

08 

365.763 

2.56320 

08 

277.298 

2.44295 

08 

223.305 

2.34890 

09 

363.828 

2.56090 

09 

276.185 

2.44120 

09 

222.583 

2.34749 

10 

361.914 

2.55861 

10 

275.080 

2.43946 

10 

221.865 

2.34609 

11 

360.020 

2.55633 

11 

273.985 

2.43773 

11 

221.152 

2.34469 

12 

358.145 

2.55406 

12 

272.898 

2.43600 

12 

220.444 

2.34330 

13 

356.290 

2.55180 

13 

271.820 

2.43428 

13 

219.740 

2.34191 

14 

354.454 

2.54956 

14 

270.750 

2.43257  1 

14 

219.041 

2.34053 

15 

352.637 

2.54733 

15 

269.689 

2.43086 

15 

218.346 

2.33915 

16 

350.838 

2.54511 

16 

268.636 

2.42916  j 

.16 

217.656 

2.33777 

17 

349.058 

2.54290 

17 

267.591 

2.42747  1 

17 

216.969 

2.33640 

18 

347.295 

2.54070 

18 

266.555 

2.42579 

18 

216.288 

2.33503 

19 

345.550 

2.53851 

19 

265.526 

2.42411 

19 

215.610 

2.33367 

20 

343.823 

2.53633 

20 

264.505 

2.42243 

20 

214.937 

2.33231 

21 

342.113 

2.53417 

21 

263.492 

2.42077 

21 

214.268 

2.33096 

22 

340.420 

2.53202 

22 

262.487 

2.41911 

22 

213.603 

2.32961 

23 

338.743 

2.52987 

23 

261.489 

2.41745 

23 

212.942 

2.32826 

24 

337.084 

2.52774 

24 

260.499 

2.41581 

24 

212.285 

2.32692 

25 

335.440 

2.52561 

25 

259.517 

2.41417 

25 

211.632 

2.32558 

26 

333.812 

2.52350 

26 

258.542 

2.41253 

26 

210.984 

2.32425 

27 

332.200 

2.52140 

27 

257.574 

2.41090 

27 

210.339 

2.32292 

28 

330.603 

2.51931 

28 

256.613 

2.40928 

28 

209.698 

2.32159 

29 

329.022 

2.51722 

29 

255.660 

2.40766 

29 

209.061 

2.32027 

30 

327.455 

2.51515 

30 

254.713 

2.40605 

30 

208.428 

2.31896 

31 

325.904 

2.51309 

31 

253.774 

2.40445 

31 

207.799 

2.31764 

32 

324.367 

2.51104 

32 

252.841 

2.40285 

32 

207.174 

2.31634 

33 

322.845 

2.50899 

33 

251.916 

2.40126 

33 

206.552 

2.31503 

34 

321.337 

2.50696 

34 

250.997 

2.39967 

34 

205.934 

2.31373 

35 

319.842 

2.50494 

35 

250.085 

2.39809 

35 

205.320 

2.31243 

36 

318.362 

2.50292 

36 

249.179 

2.39651 

36 

204.709 

2.31114 

37 

316.896 

2.50092 

37 

248.280 

2.39494 

37 

204.102 

2.30985 

38 

315.442 

2.49892 

38 

247.387 

2.39338 

38 

203.499 

2.30856 

39 

314.003 

2.49693 

39 

246.501 

2.39182 

39 

202.899 

2.30728 

40 

312.576 

2.49496 

40 

245.621 

2.39027 

40 

202.303 

2.30600 

41 

311.162 

2.49299 

41 

244.748 

2.38872 

41 

201.710 

2.30473 

42 

309.761 

2.49103 

42 

243.880 

2.38718 

42 

201.121 

2.30346 

43 

308.372 

2.48907 

43 

243.019 

2.38564 

43 

200.535 

2.30219 

44 

306.996 

2.48713 

44 

242.164 

2.38411 

44 

199.952 

2.30093 

45 

305.632 

2.48520 

45 

241.314 

2.38258 

45 

199.373 

2.29967 

46 

304.280 

2.48327 

46 

240.471 

2.38106 

46 

198.798 

2.29841 

47 

302.940 

2.48136 

47 

239.634 

2.37955 

47 

198.225 

2.29716 

48 

301.612 

2.47945 

48 

238.802 

2.37804 

48 

197.656 

2.29591 

49 

300.295 

2.47755 

49 

237.976 

2.37653 

49 

197.090 

2.29466 

50 

298.990 

2.47566 

50 

237.156 

2.37503 

50 

196.528 

2.29342 

51 

297.696 

2.47377 

51 

236.342 

2.37354 

51 

195.968 

2.29219 

52 

296.414 

2.47190 

52 

235.533 

2.37205 

52 

195.412 

2.29095 

53 

295.142 

2.47003 

53 

234.730 

2.37057 

53 

194.859 

2.28972 

54 

293.881 

2.46817 

54 

233.932 

2.36909 

54 

194.309 

2.28849 

55 

292.631 

2.46632 

55 

233.139 

2.36761 

55 

193.762 

2.28727 

56 

291.392 

2.46448 

56 

232.352 

2.36615 

56 

193.218 

2.28605 

57 

290.163 

2.46264 

57 

231.570 

2.36468 

57 

192.677 

2.28483 

58 

288.944 

2.46081 

58 

230.794 

2.36322 

58 

192.140 

2.28362 

59 

287.735 

2.45899 

59 

230.022 

2.36177 

59 

191.605 

2.28241 

TABLES 

RADII  —  METRIC  —  (Continued) 


301 


Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

6°  00' 

191.073 

2.28120 

7°  00' 

163.804 

2.21432 

8°  00' 

143.356 

2.15642 

01 

190.544 

2.28000 

01 

163.415 

2.21329 

01 

143.058 

2.15551 

02 

190.018 

2.27879 

02 

163.029 

2.21226 

02 

142.762 

2.15461 

03 

189.496 

2.27760 

03 

162.644 

2.21124 

03 

142.467 

2.15371 

04 

188.976 

2.27641 

04 

162.261 

2.21021 

04 

142.173 

2.15282 

05 

188.458 

2.27521 

05 

161.880 

2.20919 

05 

141.880 

2.15192 

06 

187.944 

2.27403 

06 

161.500 

2.20817 

06 

141.589 

2.15103 

07 

187.432 

2.27284 

07 

161.122 

2.20715 

07 

141.299 

2.15014 

08 

186.923 

2.27166 

08 

160.746 

2.20614 

08 

141.010 

2.14925 

09 

186.417 

2.27049 

09 

160.372 

2.20513 

09 

140.722 

2.14836 

10 

185.914 

2.26931 

10 

160.000 

2.20412 

10 

140.435 

2.14748 

11 

185.413 

2.26814 

11 

159.629 

2.20311 

11 

140.150 

2.14659 

12 

184.915 

2.26697 

12 

159.260 

2.20211 

12 

139.865 

•2.14571 

13 

184.420 

2.26581 

13 

158.892 

2.20110 

13 

139.582 

2.14483 

14 

183.927 

2.26465 

14 

158.527 

2.20010 

14 

139.300 

2.14395 

15 

183.437 

2.26349 

15 

158.163 

2.19910 

15 

139.019 

2.14307 

16 

182.950 

2.26233 

16 

157.800 

2.19811 

16 

138.739 

2.14220 

17 

182.465 

2.26118 

17 

157.440 

2.19712 

17 

138.460 

2.14132 

18 

181.983 

2.26003 

18 

157.081 

2.19612 

18 

138.183 

2.14045 

19 

181.503 

2.25888 

19 

156.724 

2.19514 

19 

137.906 

2.13958 

20 

181.026 

2.25774 

20 

156.368 

2.19415 

20 

137.631 

2.13872 

21 

180.552 

2.25660 

21 

156.014 

2.19316 

21 

137.357 

2.13785 

22 

180.079 

2.25546 

22 

155.661 

2.19218 

22 

137.084 

2.13699 

23 

179.610 

2.25433 

23 

155.310 

2.19120 

23 

136.812 

2.13612 

24 

179.142 

2.25320 

24 

154.961 

2.19022 

24 

136.541 

2.13526 

25 

178.678 

2.25207 

25 

154.613 

2.18925 

25 

136.271 

2.13440 

26 

178.215 

2.25094 

26 

154.267 

2.18827 

26 

136.002 

2.13355 

27 

177.755 

2.24982 

27 

153.923 

2.18730 

27 

135.734 

2.13269 

28 

177.298   2.24870 

28 

153.580 

2.18633 

28 

135.468 

2.13184 

29 

176  842   2.24759 

29 

153.238 

2.18537 

29 

135.202 

2.13098 

30 

176.389 

'  2.24647 

30 

152.898 

2.18440 

30 

134.937 

2.13013 

31 

175.939 

2.24536 

31 

152.559 

2.18344 

31 

134.674 

2.12928. 

32 

175.490   2.24425 

32 

152.222 

2.18248 

32 

134.411 

2.12843 

33 

175.044  i  2.24315 

33 

151.887 

2.18152 

33 

134.150 

2.12759 

34 

174.600  ;  2.24204 

34 

151.553 

2.18056 

34 

133.889 

2.12674 

35 

174.159   2.24095 

35 

151.220 

2.17961 

35 

133.630 

2.12590 

36 

173.720   2.23985 

36 

150.889 

2.17866 

36 

133.371 

2.12506 

37 

173.283   2.23876 

37 

150.559 

2.17771 

37 

133.114 

2.12422 

38 

172.848 

2.23766 

38 

150.231 

2.17676 

38 

132.857 

2.12338 

39 

172.415 

2.23658 

39 

149.904 

2.17581 

39 

132.602 

2.12255 

40 

171.984 

2.23549 

40 

149.579 

2.17487 

40 

132.347 

2.12171 

41 

171.556 

2.23441 

41 

149.255 

2.17393 

41 

132.094 

2.12088 

42 

171.130 

2.23333 

42 

148.932 

2.17299 

42 

131.841 

2.12005 

43 

170.705 

2.23225 

43 

148.611 

2.17205 

43 

131.589 

2.11922 

44 

170.284 

2.23117 

44 

148.291 

2.17111 

44 

131.339 

2.11839 

45 

169.863 

2.23010 

45 

147.973 

2.17018 

45 

131.089 

2.11757 

46 

169.446 

2.22903 

46 

147.656 

2.16925 

46 

130.840 

2.11674 

47 

169.030 

2.22796 

47 

147.340 

2.16832 

47 

130.593 

2.11592 

48 

168.616 

2.22690 

48 

147.026 

2.16739 

48 

130.346 

2.11510 

49 

168.204 

2.22584 

49 

146.713 

2.16647 

49 

130.100 

2.11428 

50 

167.794 

2.22478 

50 

146.401 

2.16554 

50 

129.855 

2.11346 

51 

167.387 

2.22372 

51 

146.091 

2.16462 

51 

129.611 

2.11264 

52 

166.981 

2.22267 

52 

145.782 

2.16370 

52 

129.368 

2.11183 

53 

166.577 

2.22162 

53 

145.474 

2.16278 

53 

129.125 

2.11101 

54 

166.175 

2.22057 

54 

145.168 

2.16187 

54 

128.884 

2.11020 

55 

165.775 

2.21952 

55 

144.862 

2.16095 

55 

128.644 

2.10939 

56 

165.377 

2.21848 

56 

144.559 

2.16005 

56 

128.404 

2.10858 

57 

164.981 

2.21743 

57 

144.256 

2.15913 

57 

128.166 

2.10777 

58 

164.587 

2.21640 

58 

143.955 

2.15823 

58 

127.928 

2.10697 

59 

164.195 

2.21536 

59 

143.655 

2.15732 

59 

127.691 

2.10616 

302 


THE    RAILROAD    TAPER 

RADII  —  METRIC  —  (Continued) 


Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

jog.  rad. 

9°  00' 

127.455 

2.10536 

10°  00' 

114.737 

2.05970 

11°  00' 

104.334 

2.01843 

01 

127.220 

2.10456 

01 

114.547 

2.05898 

01 

104.177 

2.01777 

02 

126.986 

2.10376 

02 

114.357 

2.05826 

02 

104.020 

2.01712 

03 

126.752 

2.10295 

03 

114.168 

2.05754 

03 

103.864 

2.01646 

04 

126.520 

2.10216 

04 

113.979 

2.05682 

04 

103.708 

2.01581 

05 

126.288 

2.10136 

05 

113.791 

2.05611 

05 

103.552 

2.01516 

06 

126.057 

2.10057 

06 

113.604 

2.05539 

06 

103.397 

2.01451 

07 

125.827 

2.09977 

07 

113.417 

2.05468 

07 

103.243 

2.01386 

08 

125.598 

2.09898 

08 

113.231 

2.05397 

08 

103.089 

2.01321 

09 

125.370 

2.09819 

09 

113.046 

2.05326 

09 

102.935 

2.01256 

10 

125.142 

2.09740 

10 

112.861 

2.05254 

10 

102.782 

2.01192 

11 

124.916 

2.09662 

11 

112.677 

2.05184 

11 

102.629 

2.01127 

12 

•124.690 

2.09583 

12 

112.493 

2.05113 

12 

102.477 

2.01063 

13 

124.465 

2.09505 

13 

112.310 

2.05042 

13 

102.325 

2.00998 

14 

124.241 

2.09426 

14 

112.128 

2.04971 

14 

102.174 

2.00934 

15 

124.017 

2.09348 

15 

111.946 

2.04901 

15 

102.023 

2.00870 

16 

123.795 

2.09270 

16 

111.765 

2.04831 

16 

101.873 

2.00806 

17 

123.573 

2.09192 

17 

111.584 

2.04760 

17 

101.723 

2.00742 

18 

123.352 

2.09115 

18 

111.404 

2.04690 

18 

101.573 

2.00678 

19 

123.132 

2.09037 

19 

111.224 

2.04620 

19 

101.424 

2.00614 

20 

122.912 

2.08959 

20 

111.046 

2.04550 

20 

101.275 

2.00550 

21 

122.694 

2.08882 

21 

110.867 

2.04480 

21 

101.127 

2.00487 

22 

122.476 

2.08805 

22 

110.689 

2.04410 

22 

100.979 

2.00423 

23 

122.259 

2.08728 

23 

110.512 

2.04341 

23 

100.832 

2.00360 

24 

122.043 

2.08651 

24 

110.336 

2.04272 

24 

100.685 

2.00296 

25 

121.827 

2.08574 

25 

110.160 

2.04202 

25 

100.538 

2.00233 

26 

121.612 

2.08498 

26 

109.984 

2.04133 

26 

100.392 

2.00170 

27 

121.398 

2.08421 

27 

109.809 

2.04064 

27  . 

100.247 

2.00107 

28 

121.185 

2.08345 

28 

109.635 

2.03995 

28 

100.102 

2.00044 

29 

120.973 

2.08269 

29 

109.461 

2.03926 

29 

99.957 

1.99981 

30 

120.761 

2.08193 

30 

109.288 

2.03857 

30 

99.812 

1.99918 

31 

120.550 

2.08117 

31 

109.115 

2.03788 

31 

99.668 

1.99856 

32 

120.340 

2.08041 

32 

108.943 

2.03720 

32 

99.525 

1.99793 

33 

120.130 

2.07965 

33 

108.771 

2.03651 

33 

99.382 

1.99731 

34 

119.921 

2.07890 

34 

108.600 

2.03583 

34 

99.239 

1.99668 

35 

119.713 

2.07814 

35 

108.430 

2.03515 

35 

99.097 

1.99606 

36 

119.506 

2.07739 

36 

108.260 

2.03447 

36 

98.955 

1.99544 

37 

119.299 

2.07664 

37 

108.090 

2.03379 

37 

98.813 

1.99481 

38 

119.093 

2.07589 

38 

107.921 

2.03311 

38 

98.672 

1.99419 

39 

118.888 

2.07514 

39 

107.753 

2.03243 

39 

98.532 

1.99358 

40 

118.684 

2.07439 

40 

107.585 

2.03175 

40 

98.391 

1.99296 

41 

118.480 

2.07364 

41 

107.418 

2.03108 

41 

98.251 

.99234 

42 

118.277 

2.07290 

42 

107.251 

2.03040 

42 

98.112 

.99172 

43 

118.074 

2.07215 

43 

107.084 

2.02972 

43 

97.973 

.99111 

44 

117.873 

2.07141 

44 

106.919 

2.02905 

44 

97.834 

.99049 

45 

117.672 

2.07067 

45 

106.753 

2.02838 

45 

97.696 

.98988 

46 

117.471 

2.06993 

46 

106.588 

2.02771 

46 

97.558 

.98926 

47 

117.272 

2.06919 

47 

106.424 

2.02704 

47 

97.420 

.98865 

48 

117.073 

2.06846 

48 

106.261 

2.02637 

48 

97.283 

.98804 

49 

116.875 

2.06772 

49 

106.097 

2.02570 

49 

97.147 

.98743 

50 

116.677 

2.06699 

50 

105.935 

2.02504 

50 

97.010 

.98682 

51 

116.480 

2.06625 

51 

105.772 

2.02437 

51 

96.874 

.98621 

52 

116.284 

2.06552 

52 

105.611 

2.02371 

52 

96.739 

.98560 

53 

116.088 

2.06479 

53 

105.449 

2.02304 

53 

96.604 

.98500 

54 

115.893 

2.06406 

54 

105.289 

2.02238 

54 

96.469 

.98439 

55 

115.699 

2.06333 

55 

105.128 

2.02172 

55 

96.334 

.98378 

56 

115.505 

2.06260 

56 

104.968 

2.02106 

56 

96.200 

.98318 

57 

115.312 

2.06187 

57 

104.809 

2.02040 

57 

96.067 

.98257 

58 

115.120 

2.06115 

58 

104.650 

2.01974 

58 

95.933 

.98197 

59 

114.928 

2.06043 

59 

104.492 

2.01908 

59 

95.800 

.98137 

TABLES 
RADII  —  METRIC  —  (Continued) 


303 


Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

12°  00' 

95.668 

1.98077 

13°  00' 

88.337 

1.94614 

14°  00' 

82.055 

.91410 

01 

95.536 

1.98017 

01 

88.224 

1.94559 

01 

81.958 

.91359 

02 

95.404 

1.97957 

02 

88.112 

1.94504 

02 

•  81.861 

.91308 

03 

95.272 

1.97897 

03 

88.000 

1.94448 

03 

81.765 

.91257 

04 

95.141 

1.97837 

04 

87.888 

1.94393 

04 

81.668 

.91205 

05 

95.010 

1.97777 

05 

87.776 

1.94338 

05 

81.572 

.91154 

06 

94.880 

1.977f7 

06 

87.665 

1.94283 

06 

81.476 

.91103 

07 

94.750 

1.97658 

07 

87.554 

1.94228 

07 

81.380 

.91052 

08 

94.620 

1.97598 

08 

87.444 

1.94173 

08 

81.285 

.91001 

09 

94.491 

.97539 

09 

87.333 

1.94118 

09 

81.190 

.90950 

10 

94.362 

.97480 

10 

87.223 

1.94063 

10 

81.095 

.90899 

11 

94.233 

.97420 

11 

87.114 

1.94009 

11 

81.000 

.90848 

12 

94.105 

.97361 

12 

87.004 

1.93954 

12 

80.905 

.90798 

13 

93.977 

.97302 

13 

86.895 

1.93899 

13 

80.811 

.90747 

14 

93.850 

.97243 

14 

86.786 

1.93845 

14 

80.717 

.90696 

15 

93.723 

.97185 

15 

86.677 

1.93790 

15 

80.623 

.90646 

16 

93.596 

.97126 

16 

86.569 

1.93736 

16 

80.529 

..90595 

17 

93.469 

.97067 

17 

86.461 

1.93682 

17 

80.436 

.90545 

18 

93.343 

.97008 

18 

86.353 

1.93628 

18 

80.342 

.90494 

19 

93.217 

.96950 

19 

86.245 

1.93573 

19 

80.249 

.90444 

20 

93.092 

.96891 

20 

86.138 

1.93519 

20 

80.156 

1.90394 

21 

92.967 

.96833 

21 

86.031 

1.93465 

21 

80.064 

1.90344 

22 

92.842 

.96774 

22 

85.924 

1.93411 

22 

79.971 

1.90293 

23 

92.717 

.96716 

23 

85.818 

1.93358 

23 

79.879 

1.90243 

24 

92.593 

.96658 

24 

85.711 

1.93304 

24 

79.787 

1.90193 

25 

92.469 

.96600 

25 

85.605 

1.93250 

25 

79.696 

1.90144 

26 

92.346 

.96542 

26 

85.500 

1.93197 

26 

79.604 

.90093 

27 

92.223 

.96484 

27 

85.394 

1.93143 

27 

79.513 

.90044 

28 

92.100 

.96426 

28 

85.289 

1.93089 

28 

79.422 

.89994 

29 

91.977 

.96368 

29 

85.184 

1.93036 

29 

79.331 

.89944 

30 

91.855 

.96310 

30 

85.079 

1.92982 

30 

79.240 

.89894  ' 

31 

91.733 

.96253 

31 

84.975 

1.92929 

31 

79.149 

.89845 

32 

91.612 

.96195 

32 

84.871 

1.92876 

32 

79.059 

.89795 

33 

91.491 

.96138 

33 

84.767 

1.92823 

33 

78.969 

.89746 

34 

91.370 

.96080 

34 

84.663 

1.92769 

34 

78.879 

.89696 

35 

91.249 

.96023 

35 

84.560 

1.92716 

35 

78.790 

.89647 

36 

91.129 

.95966 

36 

84.457 

1.92664 

36 

78.700 

.89597 

37 

91.009 

.95908 

37 

84.354 

1.92611 

37 

78.611 

.89548 

38 

90.890 

.95852 

38 

84.251 

1.92558 

38 

78.522 

.89499 

39 

90.770 

.95794 

39 

84.149 

1.92505 

39 

78.433 

.89450 

40 

90.652 

1.95738 

40 

84.047 

1.92452 

40 

78.344 

.89401 

41 

90.533 

1.95681 

41 

83.945 

1.92399 

41 

78.256 

.89352 

42 

90.415 

1.95624 

42 

83.843 

1.92347 

42 

78.168 

.89303 

43 

90.297 

1.95567 

43 

83.742 

.92294 

43 

78.080 

.89254 

44 

90.179 

1.95511 

44 

83.640 

.92241 

44 

77.992 

.89205 

45 

90.061 

1.95454 

45 

83.540 

.92189 

45, 

77.904 

.89156 

46 

89.944 

1.95397 

46 

83.439 

.92137 

46 

77.817 

.89107 

47 

89.828 

1.95341 

47 

83.339 

.92085 

47 

77.729 

.89058 

48 

89.711 

1.95285 

48 

83.238 

.92032 

48 

77.642 

.89010 

49 

89.595 

1.95228 

49 

83  .  138 

.91980 

49 

77.556 

.88962 

50 

89.479 

1.95172 

50 

83.039 

.91928 

50 

77.469 

.88913 

51 

89.363 

1.95116 

51 

82.939 

.91876 

51 

77.382 

.88864 

52 

89.248 

1.95060 

52 

82.840 

.91824 

52 

77.296 

.88816 

53 

89.133 

1.95004 

53 

82.741 

.91772 

53 

77.210 

.88767 

54 

89.019 

1.94948 

54 

82.642 

.91720 

54 

77.124 

.88719 

55 

88.904 

1.94892 

55 

82.544 

.91669 

55 

77.039 

.88671 

56 

88.790 

1.94836 

56 

82.446 

.91617 

56 

76.953 

.88623 

57 

88.676 

1.94781 

57 

82.348 

.91565 

57 

76.868 

.88575 

58 

88.563 

1.94725 

58 

82.250 

1.91514 

58 

76.783 

1.88527 

59 

88.450 

1.94670 

59 

82.152 

1.91462 

59 

76.698 

1.88478 

304 


THE   RAILROAD    TAPER 
RADII  —  METRIC  —  (Continued) 


Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

Deg. 

Radius. 

Log.  rad. 

15°  00' 

76.613 

1.88430 

19°  00' 

60.589 

1.78239 

23°  00' 

50.159 

1.70035 

05 

76.192 

1.88191 

05 

60.326 

1.78050 

15 

49.626 

1.69571 

10 

75.776 

1.87953 

10 

60.067 

.77864 

30 

49.106 

.69113 

15 

75.364 

1.87716 

15 

59.809 

.77677 

45 

48.596 

.68660 

20 

74.957 

1.87481 

20 

59.554 

.77491 

24°  00' 

48.097 

.68212 

25 

74.554 

1.87247 

25 

59.300 

.77305 

15 

47.609 

.67769 

30 

74.156 

1.87015 

30 

59.050 

.77122 

30 

47.130 

.67330 

35 

73.762 

1.86783 

35 

58.801 

.76938 

45 

46.662 

.66896 

40 

73.372 

1.86553 

40 

58.554 

.76756 

25°  00' 

46.202 

.66466 

45 

72.986 

1.86324 

45 

58.309 

.76574 

15 

45.752 

.66041 

50 

72.604 

1.86096 

50 

58.067 

.76393 

30 

45.311 

.65620 

55 

72.227 

1.85870 

55 

57.826 

.76212 

45 

44.878 

.65203 

16°  00' 

71.853 

1.85644 

20°  00' 

57.588 

.76033 

26°  00' 

44.454 

.64791 

05 

71.483 

1.85420 

05 

57.351 

.75854 

15 

44.038 

.64383 

10 

71.117 

1.85197 

10 

57.117 

.75677 

30 

43.630 

.63979 

15 

70.755 

1.84976 

15 

56.884 

.75499  i 

45 

43.230 

.63579 

20 

70.396 

1.84755 

20 

56.653 

.75322 

27°  00' 

42.837 

.63182 

25 

70.041 

1.84535 

25 

56.425 

.75147 

15 

42.451 

.62789 

30 

69.690 

1.84317 

30 

56.198 

.74972 

30 

42.072 

.62399 

35 

69.342 

1.84100 

35 

55.973 

.74798 

45 

41.701 

.62015 

40 

68.998 

1.83884 

40 

55.749 

.74624 

28°  00' 

41.336 

.61633 

45 

68.657 

1.83668 

45 

55.528 

.74451 

15 

40.977 

.61254 

50 

68.320 

1.83455 

50 

55.308 

.74279 

30 

40.625 

.60879 

55 

67.986 

1.83242 

55 

55.090 

.74107 

45 

40.279 

.60508 

17°  00' 

67.655 

1.83030 

21°  00' 

54.874 

.73937 

29°  00' 

39.939 

.60140 

05 

67.327 

1.82819 

05 

54.660 

.73767 

15 

39.605 

.59775 

10 

67.003 

1.82609 

10 

54.447 

.73597 

30 

39.277 

.59414 

15 

66.682 

1.82401 

15 

54.236 

.73429 

45 

38.954 

.59055 

20 

66.363 

1.82193 

20 

54.026 

.73260 

30°  00' 

38.637 

.58700 

25 

66.058 

1.81986 

25 

53.819 

.73094 

30 

65.736 

1.81780 

30 

53.612 

.72926 

35 

65.427 

.81576 

35 

53.408 

.72761 

40 

65.121 

.81372 

40 

53.205 

.72595 

45 

64.818 

.81170 

45 

53.003 

.72430 

50 

64.517 

.80967 

50 

52.804 

.72267 

55 

64.219 

.80766 

55 

52.605 

.72103 

18°  00' 

63.924 

.80566 

22°  00' 

52.408 

.71940 

05 

63.632 

.80368 

05 

52.213 

.71778 

10 

63.343 

.80170 

10 

52.019 

.71616 

15 

63.056 

.79973 

15 

51.827 

.71456 

20 

62.772 

.79777 

20 

51.636 

.71295 

25 

62.490 

.79581 

25 

51.446 

.71135 

30 

62.211 

.79387 

30 

51.258 

.70976 

35 

61.935 

.79194 

35 

51.072 

.70818 

40 

61.661 

.79001 

40 

50.886 

.70660 

45 

61.389 

.78809 

45 

50.702 

.70503 

50 

61.120 

.78618 

50 

50.520 

.70346 

55 

60.853 

.78428 

55 

50.338 

.70190 

TABLES  305 

NOTATION 

A  =  Curvature  increment  of  taper  per  10  meters  of  taper. 
M  =  Degree  of  main  circular  curve. 
I  =  Length  of  taper  from  the  B.C.  of  the  taper. 
/  =  Deflection  from  tangent  at  the  B.C.  of  the  taper. 
f'j  /i>  /2j  etc.  =  Deflection  from  tangent  at  some  point  on  the 

taper. 
b  =  Deflection  from  tangent  at  the  end  of  the  taper,  C.C.  of 

main  curve,  to  the  B.C.  of  the  taper. 
o  =  Central  angle  of  the  taper  at  each  end  of  the  curve. 
d  =  Circular  curve  offset  from  tangent  to  provide  room  for 

the  taper. 

t  =  Distance  along  the  tangent,  from  opposite  the  offset  posi- 
tion of  the  B.C.  of  the  simple  curve  to  the  B.C.  of  the 

taper. 

D  =  R  +  d,  where  R  is  the  radius  of  the  main  circular  curve. 
x  and  y  are  co-ordinates  of  points  on  the  taper,  referred  to  the 

tangent  at  the  B.C.  of  the  taper  as  meridian,  and 

to  a  radial  line  at  the  B.C.  of  the  taper  as  base. 
Ic  =  Long  chord  of  the  taper,  from  the  B.C.  of  the  taper  to 

points  on  the  taper, 
s  =  Distance  along  the  taper,  from  the  B.C.  of  the  taper  to 

the  transit,  when  an  intermediate  set-up  is  necessary. 
T  =  Semi-tangent  of  the  entire  tapered  curve,  from  P.I.  to 

B.C.  or  E.G. 
E  =  External,  the  distance  from  the  P.I.  to  a  point  on  the  curve 

radially  opposite. 

EQUATIONS 

Eqn.  No. 

T  =  t  +  D  tan  \I.  (7)  Both  ends  the  same 

taper. 

Tt  =  t  +  D  tan  \  I  -  d  esc  7.  (8)  Tapered  end  —  other 

end  not  tapered. 

Tn  =  Rtan  \  I  +  <Z  csc  7.  (9)  Untapered  end — 

other  end  tapered. 

T7!  =  ti  +  Di  tan  £  7  -  (di  —  d2)  esc  7.  (10)  Both  ends  tapered, 

with  different  ta- 
pers. 

T2  =  U  +  A>  tan  |  7  -  (d*  -  di)  esc  7.  (11)  Both  ends  tapered, 

with  different  ta- 
pers. 

E  =  D  sec  %  I  -  R.  (12)  Both  ends  the  same 

taper. 


306  THE    RAILROAD   TAPER 

EMPIRICAL  EQUATIONS  DERIVED  FROM  THE 
METRIC  TABLES 


A 

(45m) 

(1  +  5)  (Z  +  10) 

(107m) 
(94m) 

20 
A 

20 
A 

3  1  (I  +  10) 

(108m) 

20 
A 

The  equations  under  paragraph  18  as  derived  from  the  metric 
taper  tables  become : 

I  =  —  (M  —  A).  (45m) 

A 

In  =  —  (M2  —  A).  (46m) 

10  M, 
Zi  =  -     -v  (47m) 


lc  =  ^  (M,  -  M!  -  A).  (48m) 

=  15(M2  +  Mi)(M2-M1-A)t 

A 
a  =  15Mi(M2rMi-A).  (51am) 


20 
A 

2M2  +  A)(M2-M1  -A) 


•'     (56m) 
+  A!-  (57m) 


TABLES 


307 


08 


-    O  •*  CO  00  t"-  <N  i—  i  »O  iO  O5  00  CO  <N  b-  CO 

rH  <M  CO  CO  CO  CO  0}  O  <*  rH  •*  i-l  CO  TH  10 
o    10  iO  iO  iO  iO  iO  IO  iO  T}H  Tj<  CO  CO  <>»  i—  i  O 


*    <M  to  CO  CO  •*  b-  iO  QC  t^  O  00  T—  (  Oi  <N      .CO 

CO  rt<  iO  iO  iO  -^  CO  i—  i  iO  CO  iQ  (N  CO  lO      .  «O 


COOO>Ot^COO<M'*i—  ICOOC100      .C^IO 
OOi—  IrHT-HO'OCO'—  l-^i—  1  CO  ^        .  IO  >O 

CO  ^  Tt<  T^l  rj<  Tfl  CO  CO  CO  <M  <M  rH  O       •  O  T-H 


. 

COCOGiO»OCO(M(MOOOO^LO      .OOCO(M 
(N  CO  CO  •*  CO  <M  i—  I  »O  (M  iO  <M  rfi      .  ^  ^f  ^ 

CO  CO  CO  CO  CO  CO  CO  C^  (N  i-l  rH  O       -OrH(N 


08 


i-  p  >-  .  ' 

«•    (M  <M  CD  iO  O  O5  CO  CO  t^  t^  i—  i  .  tO  »O  rH  C<1 

iO  O  OOO-^  CO  r-t  rt<  T-H  rj<  .r^cOCOCO 

0     (M  CO  CO  CO  CO  (M  <N  (M  rH  r-l  O  -OrH 


. 

*^c<j'ocot^»ooocoajt^    .r-icxDOt^oo 

(M  CO  CO  CO  (M  rH  >O  CO  O  CO   .  TjH  (M  <N  rH  rH 
0  (N  CN  (M  <N  <N  <M  rH  rH  rH  O   '  O  rH  <M  CO  ^ 


OB 


0     rH  C<J  (N  (M  rH  rH  rH  rH  O       '  O  rH  C^  CO  ^  >O 


qS 

o?,, 


. 

»  IO  rH  C<tt^  OO  CO  Tf  O   .  CO  CO  l>-  t^  rH  O  «O 

CO  •"*!  rf  CO  05  rH  IO  CO   .  CO  rH  IO  -^  ^  •*  "^ 

0  ^H  rH  rH  rH  rH  rH  O  O   '  O  rH  rH  (M  CO  ^  >O 


08 


^i00500CO<Mt^.CO      .O»O 

r-l  r-|  rH  rH  O  ^  <M    .  CO  O 
0  rH  rH  rH  rH  rH  O  O   '  O  rH  rH  (M  CO  ^  »O  O 


dt 


OrHOOOO       'GO  rH  <M  COCO"*  I0t^ 


OOOOO     'OO  rH 


Pi, 


OOOO       •  O  O  rH  rH  (M  CO  ^  "*  IO  CO  OO 


08 


0°0 


. 

rH^HrH        '.  rH  CO  O  CO  O  ^  CO  (M  CO  (M  <M  CO 
OOO        'OOrHrHI 


OO  •  O  O  O  rH  rH  <M  CO  CO  ^  »O  CO  l^  O5 


02 


O   '  O  O  O  rH  rH  <M  (M  CO  -^  »O  CO  t^  OO  Oi 


308 


THE    RAILROAD    TAPER 


1  O  O5  i 


CX)OOO' 


co  co  cc  cs  <N  cs  cs  <N  c<  cs  <M  cc  cq  c^  <N 


O  O  i-H  i-H-i-i  i-H<M<M<N(MCOCOCOCO^ 


TABLES 


309 


TAPER  \  METRIC  —  INTERMEDIATE  VALUES     A  = 


M 

I 

/ 

b 

o 

D 

logD 

d 

t 

X 

y 

Ic 

0  45 

20.000 

0      1 

009* 

0      t 

013 

022| 

1527.921 

3.18410 

0.022 

10.000 

0.055 

20.000 

20.000 

50 

23.333 

Oil! 

017* 

0291 

1375.141 

3.13835 

0.030 

11.667 

0.080 

23.333 

23.333 

55 

26.667 

014* 

0221 

0361 

1250.144 

3.09696 

0.041 

13.333 

0.113 

26.666 

26.667 

1  00 

30.000 

017* 

027* 

045 

1145.984 

3.05918 

0.054 

15.000 

0.153 

29.999 

30.000 

05 

33.333 

020f 

033* 

0541 

1057.855 

3.02443 

0.070 

16.667 

0.201 

33.333 

33.333 

10 

36.667 

0241 

040 

1041 

982.318 

2.99225 

0.088 

18.332 

0.259 

36.665 

36.666 

15 

40.000 

028 

047 

1  15 

916.860 

2.96230 

0.109 

19.999 

0.327 

39.998 

39.999 

20 

43.333 

032}- 

054* 

1261 

859.589 

2.93429 

0.133 

21.666 

0.407 

43.331 

43.332 

25 

46.667 

036| 

021 

1391 

809.062 

2.90798 

0.160 

23.331 

0.497 

46.662 

46.665 

30 

50.000 

0411 

HI 

152* 

764.156 

2.88318 

0.190 

24.998 

0.600 

49.994 

49.998 

35 

53.333 

0461 

20| 

206! 

723.984 

2.85973 

0.225 

26.664 

0.716 

53.325 

53.331 

40 

56.667 

051* 

301 

221! 

687.837 

2.83749 

0.263 

28.330 

0.847 

56.656 

56.663 

45 

60.000 

057 

40* 

237* 

655.139 

2.81633 

0.305 

29.996 

0.993 

59.986 

59.995 

50 

63.333 

102| 

151* 

2541 

625.423 

2.79617 

0.351 

31.662 

1.154 

63.316 

63.326 

55 

66.667 

108| 

203 

311! 

598.299 

2.77692 

0.402 

33.327 

1.332 

66.644 

66.658 

2  00 

70.000 

115 

215 

330 

573.445 

2.75849 

0.458 

34.992 

1.527 

69.972 

69.989 

05 

73.333 

121* 

2271 

3491 

550.588 

2.74083 

0.518 

36.657 

1.740 

73.299 

73.319 

10 

76.667 

128* 

2401 

4091 

529.499 

2.72387 

0.583 

38.321 

1.972 

76.623 

76.650 

15 

80.000 

135* 

254* 

430 

509.982 

2.70755 

0.654 

39.986 

2.224 

79.948 

79.979 

20 

83.333 

143 

3081 

451! 

491.872 

2.69185 

0.731 

41.651 

2.497 

83.270 

83.307 

25 

86.667 

150f 

323* 

5141 

475.020 

2.67671 

0.813 

43.314 

2.791 

86.590 

86.635 

30 

90.000 

1581 

3381 

537* 

459.303 

2.66210 

0.900 

44.977 

3.107 

89.909 

89.962 

35 

93.333 

207 

3541 

601! 

444.612 

2.64798 

0.994 

46.639 

3.446 

93.224 

93.288 

40 

96.667 

2  15| 

4111 

626! 

430.851 

2.63433 

.094 

48.302 

3.808 

96.538 

96.614 

45 

100.000 

224* 

428 

652* 

417.937 

2.62111 

.200 

49.964 

4.196 

99.849 

99.937 

50 

103.333 

233* 

4451 

7191 

405.795 

2.60831 

.313 

51.626 

4.608 

103.157 

103.259 

55 

106.667 

2421 

504 

746! 

394.361 

2.59589 

.433 

53.285 

5.047 

106.461 

106.581 

3  00 

110.000 

252* 

522* 

8  15 

383.576 

2.58385 

.560 

54.945 

5.512 

109.762 

109.900 

05 

113.333 

302* 

5411 

8441 

373.387 

2.57216 

.694 

56.605 

6.005 

113.059 

113  217 

10 

116.667 

312| 

601* 

9141 

363.750 

2.56080 

.836 

58.262 

6.526 

116.351 

116.534 

15 

120.000 

323 

622 

945 

354.622 

2.54977 

.985 

59.920 

7.077 

119.639 

119.848 

20 

123.333 

334 

6421 

1016! 

345.965 

2.53903 

2.142 

61.576 

7.658 

122.922 

123.160 

25 

126.667 

345 

7041 

10491 

337.747 

2.52859 

2.307 

63.231 

8.269 

126.199 

126.470 

30 

130.000 

3561 

7261 

1122* 

329.935 

2.51843 

2.480 

64.885 

8.911 

129.470 

129.775 

35 

133.333 

407f 

749 

1156! 

322.503 

2.50853 

2.661 

66.539 

9.586 

132.734 

133.078 

40 

136.667 

4191 

8  12 

1231! 

315.427 

2.49890 

2.851 

68.189 

10.294 

135.992 

136.380 

45 

140.000 

432 

835* 

1307* 

308.683 

2.48951 

3.051 

69.840 

11.035 

139.242 

139.678 

50 

143.000 

4441 

900 

13441 

302.249 

2.48036 

3.259 

71.489 

11.810 

142.484 

142.973 

55 

146.667 

457 

9241 

1421! 

296.107 

2.47145 

3.476 

73.135 

12.620 

145.718 

146.265 

4  00 

150.000 

510 

950 

1500 

290.240 

2.46276 

3.703 

74.781 

13.467 

148.942 

149.550 

310 


THE    RAILROAD    TAPER 


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TABLES 


311 


II 

OOOiOiCiOSO^OiOOt^cDCO 

o 

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312  THE   RAILROAD   TAPER 

TAPER  1  METRIC.  —  INTERMEDIATE  VALUES 


M 

I 

/ 

6 

0 

D 

logD 

i 

t 

X 

I 

k 

1  30 

20.000 

018! 

0261 

045 

764.009 

2.88310 

0.043 

10.000 

0.109 

20.000 

20.000 

35 

21.667 

021 

0303 

051| 

723.8102.859620.051 

10.833 

0.133 

21.666 

21  667 

40 

23.333 

023* 

034; 

0581 

687.634:2.837360.060 

11.666 

0.160 

23.333 

23.333 

45 

25.000 

0261 

039; 

105* 

654.905 

2.816180.071 

12.499 

0  191 

24.999 

25.000 

50 

26.667 

029 

044; 

1  131 

625.154 

2.795990.082 

13.333 

0.225 

26.665 

26.666 

55 

28.333 

032 

049i 

1  21* 

597.991 

2.776690.094 

14.166 

0.263 

28.332 

28.333 

200 

30.000 

035 

055 

130 

573.095 

2.75823 

0.108 

14.999 

0.305 

29.998 

29.999 

05 

31.666 

038| 

100! 

139 

550.193 

2.74052 

0.123 

15.832 

0.352 

31.664 

31.665 

10 

33.333 

041* 

106| 

1481 

529.055 

2.723500.139 

16.665 

0.403    33.330 

33.332 

15 

35.000 

045 

1  13 

158 

509.484 

2.707130.156 

17.498 

0.458!  34.996,  34  998 

20 

36.667 

048* 

1  195 

2081 

491.316 

2.691360.175 

18.331 

0.518!  36.661    36  665 

25 

38.333 

052; 

1  261 

219 

474.402 

2.67615 

0.195 

19.164 

0.583    38.327    38  331 

30 

40.000 

056; 

133! 

230 

458.620 

2.66145 

0.217 

19.997 

0.654    39.992;  39.997 

35 

41.667 

100: 

1411 

241* 

443.858 

2.64725 

0.240 

20.829 

0.730!  41.6571  41  663 

40 

43.333 

1  04i 

1  48| 

2531 

430.022 

2.63347 

0.265 

21.662 

0.812 

43.322!  43.328 

45 

45.000 

1081 

156! 

305* 

417.028 

2.62017 

0.291 

22.495 

0.900 

44.986 

44.995 

50 

46.667 

1  131 

205 

3  181 

404.801 

2.607240.319 

23.327 

0.993 

46.650 

46.661 

55 
3  00 

48.333 
50.000 

1171 
122* 

213! 
222* 

331* 
345 

393.277 
382.396 

2.59470:0.349 
2.58251  0.380 

24.160 
24.992 

1.093 
1.200 

48.314 
49.977 

48.326 
49.992 

05 

51.667 

1271 

231| 

359 

372.106 

2.570670.413 

25.824 

1.313 

51.640 

51.657 

10 

53.333 

132 

241 

4131 

362.363 

2.55914  0.449 

26.657 

1.432 

53.302 

53.321 

15 

55.000 

137^ 

250* 

428 

353.123 

2.54793  0.486 

27.489 

1.559 

54.9641  54.986 

20 

56.667 

142 

300* 

4431 

344.348 

2.537000.525 

28.320 

1.693 

56.626 

56.650 

25 

58.333 

148 

3  lOf 

459 

336.006 

2.526350.566 

29.152 

1.835 

58.286 

58.314 

30 

60.000 

153 

32H 

515 

328.064 

2.51596|0.609 

29.984 

1.984 

59.946 

59.978 

35 

61.667 

15& 

332 

53U 

320.496 

2.  50582  '0.654 

30.815 

2.141 

61.606 

61  643 

40 

63.333 

2051 

343 

5481 

313.278 

2.495930.702 

31.646 

2.306 

63.264 

63.306 

45 

65.000 

21H 

3541 

605* 

306.384 

2.486270.752 

32.477 

2.480 

64.922 

64.969 

50 

66.667 

2  171 

406 

6231 

299.794 

2.476820.804 

33.308 

2.662 

66.579 

66.633 

55 

68.333 

223* 

4  18 

ft  41} 

293.489 

2.46759 

0.858 

34.139 

2.852 

68.235 

68.294 

4  00 

70.000 

230 

430 

700 

287.452 

2.45857 

0.915 

34.969 

3.051 

69.889 

69.956 

05 

71.667 

236* 

442* 

719 

281.666 

2.449730.974 

35.799 

3.260 

71.543 

71.618 

10 

73.333 

2431 

455 

7381 

276.1152.44109    .035 

36.629 

3.477 

73.195 

73.278 

15 

75.000 

250 

508 

758 

270.7882.43263    .099 

37.459 

3.704 

74.847 

74.939 

20 

76.667 

257 

5211 

8181 

265.671  2.42434 

.166 

38.288 

3.941 

76.497 

76.600 

25 

78.333 

304 

535 

839 

260.7522.41623 

.235 

39.117 

4.187 

78.145 

78.259 

30 

80.000 

3  Hi 

548f 

900 

256.0202.40827 

.307 

39.946 

4.444 

79.792 

79.918 

35 

81.667 

318^ 

603 

921| 

251.4672.40048 

.382 

40.774 

4.710 

81.437 

81.576 

40 

83.333 

326 

6171 

9431 

247.0802.39284 

.459 

41.602 

4.987 

83.081 

83.232 

45 

85.000 

333 

631! 

1005| 

242.8542.38535 

.540 

42.430 

5.274 

84.723 

84.888 

50 

86.667 

341 

646! 

10281 

238.7792.37800 

.623 

43.257 

5.572 

86.363 

86.543 

55 

88.333 

349 

702 

1051* 

234.8482.37079 

.709 

44.084 

5.882 

88.000 

88.196 

5  00 

90.000 

357 

717i 

1115 

231.0542.36372 

.798 

44.911 

6.202 

89.636 

89.850 

05 

91.667 

405f 

7331 

1139 

227.3902.35677 

.890 

45.736 

6.534 

91.270 

91.503 

10 

93.333 

414 

7491 

12031 

223.8502.34996    .985 

46.562 

6.877 

92.901 

93.154 

15 

95.000 

422* 

805* 

1228 

220.429  2.34327  2.083 

47.387 

7.232 

94.529 

94.805 

20 

96.667 

431 

8221 

12  531 

217.12212.336702.185 

48.211 

7.599 

96.155 

96.455 

25 

98.333 

4391 

8391 

1319 

213.  92112.  33025:2.  289 

49.035 

7.977 

97.780 

98.103 

30 

100.000 

448f 

8561 

1345 

210.8252.323922.397 

49.858    8.369 

99.399 

99.751 

35 

101.667 

457f 

913! 

14114 

207.828|2.31770  2.  508 

50.681    8.772 

101.016 

101.401 

40 

103.333 

507 

9311 

14  381  204.  925|2.31159  2.622 

51.503    9.188 

102  631 

103.041 

45 

105.000 

516J 

9491 

1505* 

202.1132.305592.740 

52.325    9.617 

104.241 

104.684 

50 

106.667 

525; 

1007^ 

15331 

199.  389  2.  29970  1  2.  861 

53.145  10.059 

105.849 

106.326 

55 

108.333 

536 

10251 

1601* 

196.7482.29391 

2.986 

53.965 

10.513 

107.452 

107.969 

6  00 

110.000 

545 

1045 

1630 

194.188 

2.28822 

3.115 

54.785 

10.981 

109.052 

109.604 

15 

115.000 

6  15 

1143 

1758 

186.958 

2.27174!3.521 

57.238 

12.466 

113.828 

114.508 

30 

120.000 

6461 

1243! 

1930 

180.3502.256123.961 

59.683 

14.077 

118.563 

119.395 

45 

125.000 

7181- 

13  46| 

2105* 

174.2992.241294.436 

62.119 

15.817 

123.252 

124.263 

7  00 

130.000 

752^ 

1452* 

2245 

168.751 

2.227254.947 

64.546 

17.689 

127.890 

129.108 

15 

135.000 

827i- 

16001 

2428 

163.658 

2.213945.495 

66.961 

19.698 

132.471 

133.929 

30 

140.000 

903i 

17111 

2615 

158.979 

2.20134 

6.081 

69.365 

21.850 

136.990 

138.718 

TABLES 


313 


I-H  O  OJ  CX)  «0  TH  (M      ; 


oil 


Oi  rH  OO  O  l>  OO  lO  t>-  TH  O  CO  »O 

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08 


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GO  O5  O5  Oi  O5  00  l>  CO  lO  CO  <N 


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CO  t^  <N  CO  00  Oi  iO  U5  T-H      -< 
iO  T-I  (M  i—  (  Tti  O  rH  O  •*      • 

i—  i    !  1-1  -^i.  co  os 


03 


rt<  10  to  TJ<  TF  co  <N  I-H    ;i-n 


. 

iO  00  CO  OS  t^  O  OO   .  O  *>•  QO  iO  t^  -^ 
•«^  iQ  iO  CO  O  C<l  i—  i   .  CO  i—  (  i—  i  CO  O  iO 

CO  CO  CO  CO  CO  <M  rH   •  ,-1  CO  iO  l>  O  <N 


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08 


o  ooo    ;o  rH  co  Tfi  «o  oo  o  co  co  O5 


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O       'OrH  rHCOTfH  CO  00  O  <M  IO  GO  rH 


314 


THE    RAILROAD   TAPER 


>  ia  c<i  GO  Oi  o  i— IC^I-HOOOOOO 

)OiOOTfll^iOCOOOiOI>OOO5 
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i  IO  t*^  CO  CO  CO  i""l  00  O^  ^t* 
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cqco^»ocoj>ooo5Or-«Nco 


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TABLES 


315 


--<<Nc<3'<j<io<ot-ooo>      oso^i-KMeacocOTtt^i^ 


<Mi-ioasoot-«oe<9 


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I  <N  (N  ( 


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5  Sis  §  sis  s  sf? 


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316 


THE   RAILROAD   TAPER 


CO'^OOOOC 


CO  O  rf<  Tf  CO  <M  <M  i— i  < 


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SCOCOCO 
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i  ^  TJ<  in  «D  »c  «s  10  «5  co  o  co  «o     t~  oo  oo  a  o  o  ~«  eo 


OJ  0 


TABLES 


317 


p8 


T-H  O5  1>-  >o  co 


ot^ioc^iot^-^oc^oi^-io    !  o 

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T-i      .  ^-i  CO  iO  00  rH 


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O       .  O  rH  (N  "*  CO  OO  O  CO  O  O  -^ 


r-Tl  r-'C-l  r-ICl 


318 


THE   RAILROAD   TAPER 


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319 


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327 


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331 


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OOI>«>O'^t|CO'-HOOiOOt>-CO  iO^COC^r-iOOiOOt~>- 

,_,  ,_,  ^H  ^H  rH  r-t  T-l  O  O  O  O  O  O  O  O  O  O  Oi  OS  Oi 

C<i(N(N(NC^C^C^<NC^<Nci 

^H  i— i  O3  rH  CO  rt<  rt*  TH  (N  1^  00  CO  <N  CO  CO  O  >O  Oi  (N  "*  ^  i— i 

i— i  <M  CO  CO  <N  -^i  00  •<*  <N  rH  <N  to  Oi  ^  O  00  CO  iO  CO  t^  Oi  CQ 

cooocoosioi— it^r^^Hooo  c^oit^ioc^ooocd^c^i-H 

»0  -^  Tt<  CO  CO  CO  <N  (M  (N  T-H  ^H  T-H  OOOOOOSOiOSOiOi 


i— i  co  co  Oi  <M  co  »-H  co  cq  oo  »o  <N 
-KNCO^OT-HCO^O^CO  iC 
(N<M  (MiMCOCOCOCO  ^^r^  ^  »O  »O  iO  CO  CO  CO  t^  t>  Ir^  00 


i-<  i— 1 1— i  I-H  C^  C^  C^  (M  C<l  <M  CQ   CO  CO  CO  CO  CO  CO  "tf  TJH  -^  T^  T*< 


336 


THE   RAILROAD   TAPER 


05  05  J>*  C^l  '^  CO  05  C^  ^^  CO  *O  "^  CO  05  05  CO  CO  00  ^"i   CO  C^l  t^"  ^O  *O 

COOt^-'^OCO'-Hf^C^  Ot^-C^iOOCOOi-HOCO   00  CO  05  00  C^ 

|>»  J>»  CO  CO  CO  *O  *O  "^  '^  CO  *"^  ^2  OO  CO  "^  ^H  0i  CO  C^    00  ^f  05  "^  05 

3^3h$$W3  ss?ig§^8SS§^o    ^^^^^ 

OJC^^OOiCOOO^OOS  Oitv-COOO^fi-H'—  i^fi—HC^         OOOOOOCO 

T—  (COi—  (t^-iOCOPOT^iO  Cs^Tt1'—  idO5r-  lOOOOOi—  i         Oi^COOOd 
CO  ^H  O5  t^»  t^*  CO  t^»  t^*  O5        ^^  CO  CO  O^  CO 

<Nco^      ^^ 

C^ 

^COiO 

t~--iOCOO1OO5Oi  OiOC^COr—  i 

CO  -**1  iO  CO  1>-  t>-  00  O  CO  iOt>-  O  < 

^T-J^^T-HT-JT-H  C^fMC^C^eO 

GO  OO  *-O  CO  T—  4  1>-  CO  C^  CO  CO  OS  00  CO  CO 

o'     OS  O5  O^  OS  O5  OS  OS  OS  GO  OO  OO  OO  OO  00  00  00  00  OO  t^»   Is*  t>» 

i—  I  1C  CO  CO  O  CO  OO  CO  CO  t^  rt<  lO  OS  i—  I  lO  CO  I>  O  ^O    OO  -^  OS  OS  O 

COI>  O  OS  OS  rf  ••*  O  O  •<*  iO  I-H  T—  i  iO  O  t^  TJH  <M  00   CO  I>  OO  !>•  -^ 

to  OS  -^  OS  »O  C^J  OS  l>  >O  C<|  i—  i  CN  -^  1>  C^  (N  ^  CN  O   O  O  T—  i  CO  CO 

oJ  t>^  co  Tt<  co  CN  o  os  06  co  ^  IN  o  06  i>i  to  -*  co  c<i      i-looio6b- 

00  00  00  00  00  00  00  t^  t-  t^  t^  t^.  t^  CO  CO  CO  CO  CO  CO        cOcO»Ototo 

ocooco  1-1  ioco  c3oo      totoiooo 

000000=000^,-, 
Hf   nNmHi   Hf»'HNH-*^lP» 

§Sg^^2^§§S 

to  to  tO  lO  1C  CO  CO  CO  CO  1>  l>-  OO  00  OS  O  O  i—  I  i-<  C^   CO  CO  T^  to  »O 


<M  -*  CO  00  O  <N  ^  CO  00  O        <N  -^  CO  00  O 

10  to  to  to  co  co  co  co  co 


TABLES 


TAPER  3|  METRIC.    CURVATURE  INCREASING  3°  00; 
EVERY   10  M.  A  =  3 


Transit 
at 

Deflections  to 

B.C. 

3°  00' 

C.C. 

6°  00' 

C.C. 

9°  00' 

C.C. 

12°  00' 

C.C. 

.15°  00' 

C.C. 

18°  00' 

C.C. 

21°  00' 

C.C. 

24°  00' 

I 

0                10 

20 

30 

40 

50 

60 

70 

0 
10 
20 
30 
40 
50 
60 
70 

O          / 

0  45' 
237| 
530 
922i 
14  15 
2007^ 
2700 

045 

'l  30 
407| 
745 
1222£ 
1800 
2437^ 

152| 
130 

2  15 
537| 
1000 
1522| 
2145 

330 
322i 
215 

300 
707| 
12  15 

18221 

5  37| 
545 
452| 
300 

815 
837| 
800 
622^ 
345 

4  30 
1007^ 

11221 
1200 
1137^ 
1015 
752| 
430 

5  15 

1500 
15521 
1545 
14  37£ 
1230 
922| 
5  15 

345 

837£ 
1430 

TAPER  3|  METRIC.    CURVATURE  INCREASING  3°  00' 
EVERY   10  M.— (Continued)  A  =  3 


M 

1 

0 

D 

log  D 

d 

t 

X 

V 

lc 

600 

10 

1  30 

191  .  138 

2.28135 

0.065 

4.998 

0.131 

9.999 

10.000 

900 

20 

430 

127.716 

2.106250.261 

9.986 

0.654 

19.986 

19.997 

1200 

30 

900 

96.320 

1.983720.652 

14.951 

1.83029.916 

29.972 

V1500 

40 

1500 

77.912 

1.89160 

1.29919.868 

3.910 

39.697 

39.889 

1800 

50 

2230 

66.186 

1.82077 

2.26224.703 

7.129 

49.166 

49.680 

2100 

60 

3130 

58.471 

1.766943.597 

29.405 

11.684 

58.076 

59.240 

2400 

70  4200 

53.451 

1.727965.354 

33.906 

17.709 

66.090 

68.421 

| 

1 

338 


THE   RAILROAD   TAPER 


CO  -^  CO  i— i  OS        00  CO  Tt<  CO  i— i  OS        OOcOr^C^^H 

rij 

CO  CO  CO  CO  to  CO        O  t^»  CO  OO  CN  CO        OSr- ii— lOOO^fCS 
T— (  T}H  1>»  CO  CO  CO         OS  i—*  ^  CO  OS  T— (         CO  CO  OO  CO  *""•<  CO  ^f 

t>.  iO  CO  CN  i— i  l>- 

,  o  irj  i— i  t^  rJH  j-* 
OS  O  i— I        W  CO  ^  to  l>-  00        O5 

Or- ICO^»O1>.        OO  O  i-H  C^  CO  tO        CO  !>.  OO  OS  O  ^-H  Ofl 
THOOC^cOO^t1        OOCOI>»'-H»OO5        COt--'— itOOTt^OO 

•^H        to  iO 
OOC5OOOO       OOOOOO       OOOOOOO 

•^  (M  O  OS  t^  co  TJH 

os  os  os  oo  oo  oo  oo 


"*  Tt<  iO  iO  »O  CO  CO 


OS  OS  O  O  T-H  1-1  (M 


t^O5(NiOOO(McO       O 
CO^OT-KNr^to        1-1 

(M  <N  CO  CO  CO  CO  CO 


»O  CO  CO  CO  CO  l>  t— 


i  O  i— i  C^  co       cOTfitOO 
1-1  T-I  iM  (N  <M  <N  iM        <M(NCOCOCOCO        CO  CO  CO  "tf 


t^»  CO  CO  t1**  CO   CO  t^*  CO  CO  I>*  CO   CO  I>*  CO  CO  !>•  CO  CO 

COOCOCOO        COCOOCOCOO        COCOOCOCOOCO 
COtOCOi-nO        OO  CO  to  CO  T— i  O        OOCOtOCO'-HOOO 


TABLES 


339 


OS  CO  t>.  i 

r-t  ^   CO  I 

•^  <M  O  i 


rH  CO  b-  CM  ' 

i— I   00  Tt<  O  ' 


JS  OS  OS  00  CO  Tt<  O  to  00  rH  CM 

^JOO^pCOCM  OOCOOO^^ 

cOt^-OOOSOs        rH  CO  •<*  CO  00  OS        rH  CN  TF  CO  l^  OS  O  CM  CO  »O  OO 

CO  CO  CO  CO  CO         Tt<  Tfi  Tfi  "tf  rti  rJH         to  to  to  iO  iO  iO  CO  CO  CO  CO  CO 


CM  CO  CO  i— I  t» 


00  OS  CO  OO  CO  O  l^  "f  CO 
to  i"~t  CO  OS  T-H  I-H  00  ^  t^* 
CO  CO  i—4  CO  C^\  !>•  r-n  CO  CO 


CO  t^  OO  OO  OS        i— i  CN  Th  CO  b-  OS        O  CM  CO  to  CO  OO        OS  O  CM  CO  CO 
CO  CO  CO  CO  CO        TjH  -^  Tti  "*  Tt<  T^         to  iO  to  iO  to  to        to  CO  CD  CO  CO 


8i—i  o  CD  o 
OS  OS  OS  r—1 

COCOCOCOCO        T}HTtiiO»OCOI>.        1^0005001-1        (MCO^»Ot^ 


i  TfH   CO  ^H  ^^  C*^  Ot>  O^  ' 

i  00  CO  O^  *O  T-H 


ii— I  OO   i— I  CM  OS  Tf  to  CO 

i  co  co   oo  os  os  co  co  co 


t^  CO  OO  CO  ' 
T— i  os  CO  -^ 


OO  00  OS  OS  OS   O  T-H  CM  CO  CO 


O 
(M  ( 


' OO  OS       O  O  i— i  <M  CO 
(N  CQ        CO  CO  CO  CO  CO 


T-H  CO  1"™*  "^  OS  CO  CM  GO  CM  l"^  CM  !>•  CO  CO  !>•  CD  !>•   OS  CO  OS  00 

"53     CO  *~^  '"H  CM  CM  ^f*  *O  b*  OS  CO  CM  "^  CO  00  '"^  CO  to   GO  T—I  "^  t^* 

I  i-H  rH  i— I  r-t  i— I  T-I  T-|  rH  CM  CM  CM  CM  CM  CO  CO  CO 

~^>  "^  CO  OS  CO  ^t^  OS  OO  CO  CO  t^*  CO  CO  OS  T-H  00  OS         "^f  CO  ( 

T— (  r- 1  O  OS  OS  t^-COtO^fCOCM  i— (  O  OS  OO  l^-  CO        tOtO^COCM 

OS  OS  OS  00  OO  OO  GO  00  00  00  OO  OOOOI>i>t^l>        l>t^l>I 

SCO  ^H  OO  CM  CM  to  00  t^.  CO  '  _ 

CO  rH  CO  rH  OS  CO  CM  IO  rH  OO  CO  tO  CO  OS  ^  l>- 

rHl>.TtlCMOS  tO^f^ftOOOrH  COON1 

Q  .     . 

CO  rH  CO  OS  !>•  tO  CO  rH  OS  tN*  CO  Tf  CO  rH  CO  OS  00 

sssfs  cf^I§l8  s^fsss  fsiss 

iO 

0    t^  OO  00  OS  OS  O  O  rH  CM  CO  Tt|  tpcOi>.oqOSO        rHCMcOT^t^ 

i  CM        CM  CM  CM  CM  CM 


Tt<  to  iO  to  to   CO  CO  CO  l>  !>•  00   OO  OS  OS  O  O  i— (   i— (  C^  CO  CO  iO 


CO 


>  t^  CO  i 

'  CO  CO  ' 
'  CO  CO  ' 


CO  CO  CO  CO  ^    -t1  "*  -^  ^  rt<  to    to  iO  to  iO  to  CO   COCOcOCO 


88888 


340  THE    RAILROAD    TAPER 

TAPER    3£    METRIC.    CURVATURE  INCREASING  4°  00' 
EVERY   10  M.  A  =  4 


Transit 
at 

Deflections  to 

B.C. 

4°  00' 

C.C. 

8°  00' 

C.C. 

12°  00' 

C.C. 

16°  00' 

C.C. 

20°  00' 

C.C. 

24°  00' 

C.C. 

28°  00' 

I 

0 

10 

20 

30 

40 

50 

60 

0 
10 

20 
30 
40 
50 
60 

0            , 

100 

230 
200 

0          / 

440 
430 
300 

'400 
930 
1620 

730 
740 
630 
400 

'sod 

11  30 

1100 
1130 
1040 
830 
500 

600 

0            / 

1510 
1600 
1530 
1340 
1030 
600 

100 
330 
720 
1230 
1900 
2650 

200 
530 
1020 
1630 
2400 

300 
730 
1320 
2030 

TAPER  3£  METRIC.    CURVATURE  INCREASING  4°  00' 
EVERY  10  M.  —  (Continued)  A  =  4 


M 

l 

o 

D 

log£> 

d 

i 

X 

y 

lc 

800 

10 

200 

143.443 

2.15668 

0.087 

4.996 

0.175 

9.999 

10.000 

1200 

20 

600 

96.016 

1.98234 

0.348 

9.974 

0.872 

19.974 

19.993 

1600 

30 

1200 

72.719 

1.86165 

0.866 

14.912 

2.437 

29.851 

29.951 

2000 

40 

2000 

59.309 

1.77312 

1.721 

19.767 

5.195 

39.463 

39.804 

2400 

50 

3000 

51.083 

1.70828 

2.986 

24.477 

9.432 

48.526 

49.434 

2800 

60 

4200 

46.067 

1.66339 

4.731 

28.957 

15.347 

56.616 

58.66^) 

TABLES                                                    341 
TAPER  3|  METRIC.     INTERMEDIATE  VALUES       A=4 

M 

I 

/ 

6 

o 

D 

log  D 

d 

t 

X 

V 

Ic 

1200 

20.000 

230 

330 

600 

96.016 

1.98234 

0.348 

9.974 

0.872 

19.974 

19.993 

15 

20.625 

237 

342 

619 

94.095 

1.97357 

0.372 

10.284 

0.941 

20.596 

20.617 

30 

21.250 

244 

353* 

638* 

92.252 

1.96498 

0.397 

10.594 

1.013 

21.217 

21.241 

45 

21.875 

251J 

407 

658i 

90.484 

1.95657 

0.423 

10.904 

1.089 

21.838 

21.865 

1300 

22.500 

258! 

420 

718! 

88.787 

1.94835 

0.450 

11.214 

1.169 

22.458 

22.488 

15 

23.125 

306i 

433J 

739* 

87.155 

1.94029 

0.478 

11.523 

1.252 

23.077 

23.111 

30 

23.750 

314 

447 

801 

85.587 

1.93241 

0.508 

11.833 

1.339 

23.696 

23.734 

45 

24.375 

322 

500! 

822! 

84.078 

1.92468 

0.538 

12.142 

.430 

24.315 

24.357 

1400 

25.000 

330 

515 

845 

82.625 

1.91711 

0.570 

12.450 

.525 

24.933 

24.979 

15 

25.625 

338i 

529* 

907! 

81.225 

1.90969 

0.602 

12.759 

.624 

25.550 

25.602 

30 

26.250 

346* 

544* 

931 

79.876 

1.90242 

0.636 

13.067 

.727 

26.167 

26.224 

45 

26.875 

355 

559* 

954* 

78.575 

1.89528 

0.671 

13.375 

.834 

26.783 

26.846 

1500 

27.500 

403! 

615 

1018! 

77.321 

1.88830 

0.708 

13.683 

.945 

27.398 

27.467 

30 

28.750 

421* 

647 

1108* 

74.940 

1.87471 

0.784 

14.298 

2.182 

28.626 

28.710 

1600 

30.000 

-  440 

720 

1200 

72.719 

1.86165 

0.866 

14.912 

2.437 

29.851 

29.951 

30 

31.250 

459 

754* 

1253* 

70.643 

1.84906 

0.953 

15.524 

2.710 

31.072 

31.190 

1700 

32.500 

518! 

830 

1348! 

68.700 

1.83696 

1.045 

16.135 

3.002 

32.288 

32.427 

30 

33.750 

539 

907 

1446 

66.879 

1.82529 

1.143 

16.745 

3.315 

33.499 

33.662 

1800 

35.000 

600 

945 

1545 

65.170 

.81405 

1.246 

17.353 

3.648 

34.704 

34.895 

30 

36.250 

621* 

1024* 

1646 

63.567 

.80323 

1.356 

17.959 

4.002 

35.904 

36.126 

1900 

37.500 

643! 

1105 

1748! 

62.060 

.79281 

1.471 

18.563 

4.377 

37.097 

37.355 

30 

38.750 

706* 

1147 

1853* 

60.643 

.78278 

1.593 

19.166 

4.775 

38.284 

38.581 

2000 

40.000 

730 

1230 

2000 

59.309 

.77312 

1.721 

19.767 

5.195 

39.463 

39.804 

30 

41.250 

754 

13  14* 

2108* 

58.053 

.76382 

1.855 

20.365 

5.639 

40.633 

41.023 

2100 

42.500 

8  18! 

1400 

2218! 

56.870 

.75488 

1.996 

20.961 

6.106 

41  .  794 

42.238 

30 

43.750 

844 

1447 

2331 

55.755 

.74628 

2.143 

21.554 

6.598 

42.945 

43.449 

2200 

45.000 

910 

1535 

2445 

54.706 

.73804 

2.298 

22.145 

7.114 

44.086 

44.656 

30 

46.250 

936* 

1624* 

2601 

53.717 

.73011 

2.459 

22.733 

7.655 

45.215 

45.858 

2300 

47.500 

1003! 

1715 

2718! 

52.787 

.72253 

2.628 

23.317 

8.222 

46.332 

47.056 

2400 

50.000 

1100 

1900 

3000 

61.083 

.70828 

2.986 

24.477 

9.432 

48.526 

49.434 

2500 

52.500 

1158! 

2050 

3248! 

49.577 

.69528 

3.375 

25.623 

10.749 

50.660 

51.787 

2600 

55.000 

1300 

2245 

3545 

48.248 

.68348 

3.794 

26.753 

12,173 

52.725 

64.111 

2700 

57.500 

1403! 

2445 

3848! 

47.083 

.67286 

4.246 

27.865 

13.705 

54.714 

56.403 

2800 

60.000 

1510 

2650 

4200 

46.067 

.66339 

4.731 

28.957 

15.347 

56.616 

58.660 

342 


THE    RAILROAD    TAPER 


TAPER    3f    METRIC.    CURVATURE    INCREASING    5°  00' 
EVERY   10  M.  A  =  5 


Transit 
at 


0 
10 
20 
30 
40 
50 


Deflections  to 


B.C. 

5°  00' 

C.C. 

10°  00' 

C.C. 

15°  00' 

C.C. 

20°  00' 

C.C. 

25°  00' 

C.C. 

30°  00' 

0 

10 

20 

30 

40 

50 

1  15 

1  15 

307^ 
230 

0          / 

550 
537| 

0          1 

922| 
935 

1345 
1422| 

422^ 
910 
15371 
2345 

230 
652£ 
1255 
20371 

'345" 
922^ 
1640 

345 

"SOO" 
11521 

807£ 
500 

6  15 

1320 
1037| 
615 

TAPER  3|  METRIC.    CURVATURE    INCREASING    5°  00' 
EVERY   10  M.  —  (Continued]  A  =  5 


M 

I 

o 

D 

log  D 

d 

t 

X 

y 

lc 

1000 

10 

230 

114.847 

2.06012 

0.110 

4.993 

0.218 

9.998 

10.000 

1500 

20 

730 

77.048 

1.88676 

0.435 

9.960 

1.090 

19.960 

19.990 

2000 

30 

1500 

58.667 

1.76839 

1.079 

14.863 

3.041 

29.767 

29.922 

2500 

40 

2500 

48.338 

1.68429 

2.136 

19.638 

6.465 

39.164 

39.694 

3000 

50 

3730 

42.331 

1.62666 

3.694 

24.192 

11.675 

47.713 

49.120 

TABLES 


343 


'  1^  CO  OS  Tt<  OO  i— ICO"*' 


^t^  C^  ^^   "^  ^2  *O  OQ  ^^ 


»— i  O  t^  tM  to  CO  Tf  OS  O  1--   T-H  T-H  to  CO  O  00  CO -*    OO  "*  00  CO  CO 
»O  "f  C^  r-t  OS  b-  to  CS  O  CO   CO  OS  Tt^  OS -^  t^.  CO  CO   »O  T-I  (M  OS  T-H 

os  os  os  os  oo  oo  oo  oo  oo  b-      t^ - 


00  >O  iO  00  »O  CO  i— I    T— it>-tX)T^ 
i— I  O  O  i— I  Tt<  OO  Tf<    i— I  O5  O5  i— I 


t^  CO  CO  O  >O 
iO  <N  CO  OO  t^ 
CO  CO  CO  ^  CO 


t^-OC^COCOC^IOt^-CO        t^-OC^COT— (OOt^-OO        OC^TfiiOO^ 

Tt^  CO   C^   »™H  (O  OS  t>*  CO  ^  CO  rH  OS  t^»  TfH  OS  CO  !>•  OS  C^  <Q  OS 

OS  T^H  OS  TT  OS  T^  OO  CO  OO   CO  OO  CO  t>»  C^  t"^  CO  CO   to  Tt^  "^  CO  T-H 


i  >O  O  O5  <M  t^  CO  O  . 
i  IQ  LO  CO  t^  t^  OO  O5 


•*  Tf<  001^ 

CO  »O  Tt<  Ttl 


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TABLES 


351 


CONVERSION  OF  ENGLISH  INCHES  INTO  CENTIMETERS. 

Ins. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

0 

0.000 

2.540     5.080 

7.620 

10.16 

12.70 

15.24 

17.78 

20.32 

22.86 

10 

25.40 

27.94      30.48 

33.02 

35.56 

38.10 

40.64 

43.18 

45.72 

48.26 

20 

50.80 

53.34 

55.88 

58.42 

60.96 

63.50 

66.04 

68.58 

71.12 

73.66 

30 

76.20 

78.74 

81.26 

83.82 

86.36 

88.90 

91.44 

93.98 

96.52 

99.06 

40 

101.60 

104.14    106.68 

109.22 

111.76 

114.30 

116.84 

119.38 

121.92 

124.46 

50 

127.00 

129.54    132.08 

134.62 

137.16 

139.70 

142.24 

144.78 

147.32 

149.86 

60 

152.40 

154.94    157.48 

160.02 

162.56 

165.10 

167.64 

170.18 

172.72 

175.  96 

70 

177.80 

180.34    182.88 

185.42 

187.96 

190.50 

193.04 

195.58 

198.12 

200.  &6 

80 

203.20 

205.74 

208.28 

210.82 

213.36 

215.90 

218.44 

220.98 

223.52 

226.06 

90 

228.60 

231.14 

233.68 

236.22 

238.76 

241.30 

243.84 

246.38 

248.92 

251.46 

100 

254.00 

256.54 

259.08 

261  .62 

264.16 

266.70 

269.  24!  271.  78 

274.32 

276.  W 

CONVERSION  OF  CENTIMETERS  INTO  ENGLISH  INCHES. 

Cm. 

0 

1 

2 

3 

4 

5 

6 

.7 

8 

9 

Ins. 

Ins. 

Ins 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

0 

0.000 

0.394 

0.787     1.181 

1.575 

1.969 

2.362 

2.756 

3.150 

3.543 

10 

3.937 

4.331 

4.742J    5.118 

5.512 

5.906 

6.299 

6.693 

7.087 

7.480 

20 

7.874 

8.268 

8.662     9.055 

9.449 

9.843 

10.236 

10.630 

11.024 

11.418 

30 

11.811 

12.205 

12.599    12.992 

13.386 

13.780 

14.173;14.567 

14.961 

15.356 

40 

15.748 

16.142 

16.530    16.929 

17.323 

17.717 

18.11i;i8.504 

18.898 

19.292 

50 

19.685 

20.079 

20.473    20.867 

21.260 

21.654 

22.04822.441 

22.835128.229  * 

60 

23.622 

24.016 

24.410    24.804 

25.197 

25.591 

25.98526.378 

26.772 

27.166 

70 

27.560 

27.953 

28.347    28.741 

29.134 

29.528 

29.92230.316 

30.709 

31.103 

80 

31.497 

31.890|  82.884'  32.678 

33.071 

33.46533.85934.253 

34.646 

35.040 

90 

35.434 

35.8271  36.221    36.615 

37.009 

37.40237.79638.190 

38.58338.977 

100 

39.370|  39.7641  40.158    40.552 

40.945 

41.33941.73342.126 

42.520l42.914 

CONVERSION  OF  ENGLISH  FEET  INTO  METERS. 

Feet. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

0 

0.000 

0.3048 

0.6096 

0.9144 

1.2192 

1.5239 

1.8287 

2.1335 

2.4383 

2.7431 

10 

3.0479 

3.3527 

8.6575 

3.9623 

4.2671 

4.5719 

4.87675.1815 

5.4863 

5.7911 

20 

6.0959 

6.4006 

6.7055 

7.0102 

7.3150 

7.6198 

7.92468.2294 

8.5342 

8.8390 

30 

9.1438 

9.4486 

9.7534 

10.058 

10.363 

10.668 

10.97211.277 

11.582 

11.887 

40 

12.192 

12.496 

12.801 

13.106 

13.411 

13.716 

14.02014.325 

14.630 

14.935 

50 

15.239 

15.544 

15.849 

16.154 

16.459 

16.763 

17.06817.373 

17.678 

17.983 

60 

18.287 

18.592 

18.897 

19.202 

19.507 

19.811 

20.11620.421 

20.726 

21.031 

70 

21.335 

21.640 

21.945 

22.250 

22.555 

22.859 

23.16423.469 

23.774 

24.079 

80 

24.383 

24.688 

24.993 

25.298 

25.602 

25.907 

26.21226.517 

26.822 

27.126 

90 

27.431 

27.736 

28.041 

28.346 

28.651 

28.955 

29.26029.565 

29.870 

30.174 

100 

30.479 

30.784 

31.089 

31.394 

31.698 

32.003 

32.30832.613 

32.918 

33.222 

CONVERSION  OF  METERS  INTO  ENGLISH  FEET. 

Met. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Feet. 

Feet. 

Feet,  i  Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

0 

0.000 

3.2809 

6.5618    9.8427 

13.123 

16.404 

19.685 

22.966 

26.247 

29.528 

10 

82.809 

36.090 

39.371    42.651 

45.932 

49.213 

52.494 

55.775 

59.056 

62.337 

20 

65.618 

68.899 

72.179    75.461 

78.741 

82.022 

85.303 

88.584 

91.865 

95.146 

30 

98.427 

101.71 

104.99    108.27 

111  55 

114.83 

118.11 

121.39 

124.67 

127.96 

40 

131.24 

134.52 

137.80    141.08 

144.36 

147.64 

150.92 

154.20 

157.48 

160.76 

50 

164.04 

167.33 

170.61    173.89 

177.17 

180.45 

183.73 

187.01 

190.29 

193.57 

60 

196.85 

200.13 

203.42    206.70 

209.98 

213.26216.54 

219.82 

223.10 

226.38 

70 

229.66 

232.94 

236.22    239.51 

242.79 

246.07,249.35 

252.63 

255.91 

259.19 

80 

262.47 

265.75 

269.03:  272.31 

275.60 

278.88282.16 

285.44 

288.72 

292.00 

90 

295.28 

298.56 

301.  841  305.12 

308.40 

311.69314.97 

318.25 

321.53 

324.81 

100 

328.09 

331.37 

334.65'  337.93 

341.21 

344.49347.78 

351.06 

354.34 

.357.02 

352 


THE   RAILROAD    TAPER 


TRIGONOMETRIC  AND   MISCELLANEOUS 
FORMULAS. 


SOLUTION   OF  RIGHT  TRIANGLES. 


Required. 
A,  (7,  C 


A,  C,  b 


C,  a,  c 
C,  a,  6 


Given, 
a,  6 

a,  c 
,  a 

A,c 


Formulas. 


=  cot  5  =  -  ;    6  = 


a)  (6  -  a), 


+  ca. 


SOLUTION  OF  OBLIQUE  TRIANGLES. 


Required. 
5 

Given. 
y4     7?    /r 

Formulas. 

A         CL        b 

sin  A 

i£j 

fee 

f 

a 

tir\lf4   —   7?\     •  ^~        ton  if  A   4-   7?\ 

^4 

P 

. 

a  +  o 

, 

,f      .,       ,,    ,      ,      p.nj   ^_      /(S  —  ft)(.S  —  C) 

a,  &,  c 

\          &c 

^ 

i  j          /(s  —  6)  (s  —  c) 

V      8(8  -a) 

cm  1       2^8(«-a)(«-6)(8-c) 

6c 

Area 
Area 

Arpa 

A,  6,  c 

>4     7?    r 

Area  =  */8  (s  —  a)  (s  —  6)  (s  —  c) 
Area  =  £  6c  sin  A 
Arca  _  c2  sin  ^  sin  B 

-A,  />,  C 

TABLES  353 

TRIGONOMETRIC  AND    MISCELLANEOUS 
FORMULAS. 

GENERAL  FORMULAS. 

sin  A  =  «/l  —  cos2  .4 
sin.4 


cosec  A 


cos  A  =  -—  =  vsin2^.  =  cot  A  sin  A, 
sec  A 

cosA  =1  —  2sini4^1  =  l  —ver&A. 


cosJ. 


_vl—  cos2 A  ___    sin2A 
cosA        ~1  + 

~~cotA~~      s'm2A 


cot  A  =  -1-  =  = 

tan  A      sin  yl 

cot^l  =     sin  2  ^4      _  1  + 
"~ 


sec^l  =  —  —  =the  reciprocal  of  any  expression  for  cos  A 
cos  A 

cosec  A  =  —  —  =  the  reciprocal  of  any  expression  for  sin  .4. 
sin-4 

vers^.  =  1  —  cos.4  =  2  sin2  1  A. 
versA 


354  THE   BAILROAD   TAPER 

TRIGONOMETRIC  AND   MISCELLANEOUS 
FORMULAS. 


i  j  _      inn  A     1  —  cos^i sin  A 

1  +  secu4          sinJ.          1  + 


sin  .4  I—  cos  A 

sin  2  .4  =  2  sin  .4  cos  A, 
cos  2  .4  =  cos2^.  —  sin2  .4  =  2  cos2  J.  —  1. 

_ll5a4- 
1  —  tanM 


2  cot  A 

sin  (^4  ±  7^)  =  sin  A  cosB  ±  cos  A  sin  B. 
cos  (A  ±  B)  =  cos^l  cos!?  q=  sin^l  sinB. 


1  =p  tan  ^1  tan  B 
Bin  A  +sinB  =  2  sin  £  (A  +  B)  cos|  (^L  —  B>- 
sin  A  —  sin  B  =  2  cos|  (A  +  B)  sin  $  (A  —  B). 
cos  A  +-cosB=2coal(A  +  B)cos$(A  —  B). 
costf  —  cosA  =  2sm$(A  +  B)sin^(^4  —  7^). 
sin2  ^4  —  sin2  B  =  cos2  B  —  cos'2  A  =  sin  (A  -f  73)  GIP  (^ 
cos2  -4  —  sin2  7?  =  cos(J.  +  73)  cos  (A  —  B). 


cos  A  cos  73 


TABLES 

TRIGONOMETRIC  AND  MISCELLANEOUS 
FORMULAS. 

MISCELLANEOUS   FORMULAS. 


355 


Required. 

Given. 

F 

Area  of 

Parallel  sides  =  in  and  n 

p 

Trapezoid 

Perp.  dist.  bet.  them  =p 

2(m 

Regular  Polygon 

Length  of  side  =  I 
Number  of  sides  =  n 

Tc' 

Circle 

Radius  =  r 

itr'1  [: 

Ellipse 

Semi-axes  =  a  and  b 

Ttab 

Parabola 

Base  =  b,  height  =  h 

Ibh 

Surface  of 

Radius  of  base  =  r 

Cone 

Slant  height  =  s 

Ttrs 

Cylinder 

Radius  =  r,  height  =  h 

2rtrh 

Sphere 

Radius  =  r 

litr* 

Zone 

Height  =  h 

2itrh 

Radius  of  its  sphere  =  r 

Volume  of 
Prism  or  cylinder 

Area  of  base  =  b 
Height  ==  h 

bh 

Pyramid  or  cone 

Area  of  base  =  b 
Height  =  h 

bh 
3 

Frustum  of 

Pyramid  or  cone 

! 

Area  of  bases  =  6  and  &' 
Height  =  h 

3 

Sphere                     \  Radius  =  r 

farr8 

Formulas. 


i.  1414J 


356  MISCELLANEOUS   FORMULAS 

THEORETICAL  SUPER-ELEVATION  IN  INCHES 


Degree  of 
Curve. 

Velocity  in  Miles  per  Hour. 

0     . 

$  ® 

bfl  3 

Q° 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

65 

70 

1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

0 

j 

' 
. 

. 

' 
" 
1 
1 
1 
1 
1 
1 
1 

. 

I 

I 

1 

4 

H 
H 
if 
H 
if 
U 
2 
21 

2l 
2! 
21 
3 

1 

3 

4 

it 

if 

21 

2f 

2f 
21 

si 

31 

31 
41 

a 

4| 
5 
51 

il 

2 

21 
21 
31 
3| 

41 
41 
5f 
51 
61 
6f 

7| 
81 

11 
If 

2f 
3 

4| 

51 

51 

71 

7f 
8| 
81 

If 

2| 

31 
4 

41 
5f 

71 

•j  i 

21 
31 

41 

51 
61 

il 

If 

2f 
4 
5f 
61 
8 

If 

41 
6f 
81 

2 
4 
6 

8 

2| 

71 

2| 

si 

31 
9| 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

... 

i 

j 

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